diff --git a/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/2301.05309v1.pdf.txt b/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/2301.05309v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..669cf83800d39b8c80dbdde2d254f03e182da372 --- /dev/null +++ b/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/2301.05309v1.pdf.txt @@ -0,0 +1,994 @@ +Planning Visual Inspection Tours for a 3D Dubins Airplane +Model in an Urban Environment +Collin Hague ∗, Andrew Willis †, Dipankar Maity ‡, Artur Wolek § +University of North Carolina at Charlotte, Charlotte, North Carolina, 28223 +This paper investigates the problem of planning a minimum-length tour for a three- +dimensional Dubins airplane model to visually inspect a series of targets located on the ground +or exterior surface of objects in an urban environment. Objects are 2.5D extruded polygons +representing buildings or other structures. A visibility volume defines the set of admissible +(occlusion-free) viewing locations for each target that satisfy feasible airspace and imaging con- +straints. The Dubins traveling salesperson problem with neighborhoods (DTSPN) is extended +to three dimensions with visibility volumes that are approximated by triangular meshes. Four +sampling algorithms are proposed for sampling vehicle configurations within each visibility +volume to define vertices of the underlying DTSPN. Additionally, a heuristic approach is pro- +posed to improve computation time by approximating edge costs of the 3D Dubins airplane +with a lower bound that is used to solve for a sequence of viewing locations. The viewing +locations are then assigned pitch and heading angles based on their relative geometry. The +proposed sampling methods and heuristics are compared through a Monte-Carlo experiment +that simulates view planning tours over a realistic urban environment. +I. Introduction +U +nmanned aerial vehicles (UAVs) are routinely used in applications such as visual reconnaissance, infrastructure +inspection, and aerial photography to image a series of points of interest (henceforth referred to as targets). In +three-dimensional environments (e.g., an urban city, mountainous terrain) the targets must be imaged from particular +vantage points to avoid occlusions from surrounding objects (e.g., buildings, trees). Additional requirements, such as +airspace restrictions and image resolution, further constrain the three-dimensional visibility volume from which an image +of a target may be obtained. This paper investigates the problem of planning a path to image a set of targets by flying +through their corresponding visibility volumes in minimum time. The UAV is modeled as a Dubins airplane [1, 2] and +the environment consists of extruded polygonal objects with targets located on the ground or on the surface of objects. +A. Relation to Prior Work +The view planning problem considered here is related to the Dubins traveling salesperson problem (DTSP [3]) of +constructing a minimum-time tour for a constant-speed planar Dubins vehicle model [4] to travel through a series of +planar points (with arbitrary heading). The set of points to visit can be generalized to arbitrary planar regions (e.g., +polygons) to give the DTSP with neighborhoods (DTSPN [5]) wherein the Dubins vehicle must visit at least one point in +each region/neighborhood. One application of the DTSPN is to plan visual inspection tours for an airplane to visit +planar polygonal regions at a constant altitude to image ground targets [6]. More recently, the Dubins airplane model +[1, 2] that includes additional degrees of freedom (altitude and pitch angle) was used to extend the DTSPN to three +dimensions. Planning three-dimensional Dubins tours have typically assumed that the desired viewing regions have +relatively simple geometries, such as spheres [7] or cylinders [8]. In contrast, this work admits more complex target +visibility volumes that are approximated as triangular meshes. +B. Contributions +This paper formulates a view planning problem for a 3D Dubins airplane model to observe a set of targets occluded +by objects in an urban environment. The contributions of the paper are: (1) four sampling algorithms that extend +∗Graduate student, Department of Mechanical Engineering and Engineering Science +†Associate Professor, Department of Electrical and Computer Engineering +‡Assistant Professor, Department of Electrical and Computer Engineering +§Assistant Professor, Department of Mechanical Engineering and Engineering Science, Member AIAA +1 +arXiv:2301.05309v1 [eess.SY] 12 Jan 2023 + +two-dimensional Dubins-based view planning to three dimensions with visibility volumes that have an arbitrary geometry +approximated by a triangular mesh, and (2) a heuristic approach that solves for a tour using a modified Euclidean +distance TSP (METSP) with edge costs that are lower bounds for the 3D Dubins path length and using the geometry of +consecutive viewing locations in the METSP tour to assign heading and pitch angles. The relative performance of the +algorithms are characterized through a Monte-Carlo experiment. +C. Paper Organization +The remainder of the paper is organized as follows. Section II describes the airplane motion model, the environment +model, the target visibility volumes, and states the view planning problem. Section III describes a method for +approximately computing the target visibility volumes and path planning for constant-altitude 2D tours. Section IV +introduces 3D path planning algorithms and proposes heuristics to reduce computation time. Section V describes the +results of a Monte-Carlo experiment that compares the 2D and 3D algorithms. The paper is concluded in Sec. VI. +II. Problem Formulation +This section formulates the problem of planning a minimum time path for an unmanned airplane to visually inspect +a set of targets in the presence of occluding structures. The vehicle motion model, environmental model, and target +visibility volumes are introduced, and the view planning problem is formally stated. +A. Airplane Motion Model +This work considers the three-dimensional Dubins airplane model [9, 10]: +������������ +�𝑥 +�𝑦 +�𝑧 +�𝜓 +�𝛾 +������������ += +������������ +𝑣 cos 𝜓 cos 𝛾 +𝑣 sin 𝜓 cos 𝛾 +𝑣 sin 𝛾 +𝑢𝜓 +𝑢𝛾 +������������ +, +(1) +where (𝑥, 𝑦, 𝑧) ∈ R3 is the inertial position of the airplane expressed in an east-north-up coordinate system, 𝑣 is the +vehicle’s speed, 𝜓 is the heading angle, and 𝛾 is the pitch angle (see Fig. 1). The control inputs are the turn-rate 𝑢𝜓 and +the pitch-angle-rate 𝑢𝛾. The Dubins airplane model travels in the direction it is pointed so that the pitch angle 𝛾 is +Fig. 1 +The model for a Dubins airplane flying at speed 𝑣 where (𝑥, 𝑦, 𝑧) is the inertial position, 𝜓 is the heading +angle, and 𝛾 is the pitch angle. +equivalent to the flight path angle and is constrained between a minimum and maximum angle, 𝛾 ∈ [𝛾min, 𝛾max]. The +controls are constrained such that the path curvature 𝜌min is bounded [11]: +𝜌min ≤ +1 +√︃ +𝑢2 +𝜓 cos2 𝛾 + 𝑢2𝛾 +. +(2) +2 + +Let the vehicle’s configuration be denoted 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 where 𝑄 = R3 × S2 is the configuration space. +An example 2D Dubins path (modified with a constant pitch angle to join two altitudes) and a 3D Dubins path that +join 𝒒𝑖 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) are shown in Fig. 2. The modified 2D Dubins path uses a +constant pitch angle 𝛾𝑐 that is computed from the change in altitude and planar displacement between the start and end +configurations. The modified 2D Dubins path does not satisfy the required pitch angle at the start/end configurations and +may violate pitch angle constraints along the path when the change in altitude is large relative to the planar displacement. +Instead, a 3D Dubins path can join two configurations while limiting the pitch angle along the path to within the +allowable bounds. The 3D Dubins paths are generated according to [10] by decomposing the 3D path into two decoupled +2D Dubins paths. First, a 2D horizontal Dubins path is constructed in the 𝑥𝑦 plane to join the 2D Dubins configurations +(𝑥𝑖, 𝑦𝑖, 𝜓𝑖) and (𝑥 𝑗, 𝑦 𝑗, 𝜓 𝑗) using a horizontal turn radius that is twice the minimum turn radius 𝜌h = 2𝜌min. Next, a 2D +vertical path is constructed, with vertical plane turn radius 𝜌v that is found from [10] +𝜌−2 +min = 𝜌−2 +h + 𝜌−2 +v +, +(3) +to join the 2D Dubins configurations (𝑠𝑖, 𝑧𝑖, 𝛾𝑖) and (𝑠 𝑗, 𝑧 𝑗, 𝛾 𝑗) where 𝑠𝑖 and 𝑠 𝑗 are the initial and final arc-lengths +along the Dubins path in the 𝑥𝑦 plane (where 𝑠𝑖 = 0). The turn radii, 𝜌h and 𝜌v, are iteratively varied while satisfying +(3) to meet the acceptable pitch angle constraint while minimizing the path length as described in [10]. The length of a +3D Dubins path between two configurations, 𝒒𝑖, 𝒒 𝑗 ∈ 𝑄 is denoted 𝐷(𝒒𝑖, 𝒒 𝑗) : 𝑄2 → R. +Start +End +Modified 2D +Dubins Path +3D Dubins Path +Fig. 2 +An example 3D Dubins airplane path (green) [10] joining configurations 𝒒1 = (0, 0, 0, 𝜋 +6 , 0) and 𝒒2 = +(0, 300 m, 400 m, 0, 0) is compared to a modified 2D Dubins path (red) that join the same pair of locations and +heading angles. The modified 2D Dubins path is shorter (523 m compared to 1184 m) but violates the pitch angle +constraint since a large altitude change is required over a relatively short distance. The paths are constructed +with the parameters: 𝜌min = 40 m, 𝛾min = −𝜋/12, and 𝛾max = 𝜋/9. +B. Environment +The airplane operates in an urban environment that consists of a ground plane and a collection of 2.5-dimensional +objects representing buildings or other structures. Let 𝑂 = {𝑂0, . . . , 𝑂𝑁𝑂−1} be the set of 𝑁𝑂 objects, where 𝑂𝑖 ⊂ R3 +for each 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1}. The 𝑖th object is an extruded polygon 𝑂𝑖 = {(𝑥, 𝑦, 𝑧) ∈ R3 | (𝑥, 𝑦) ∈ 𝐴𝑖 and 𝑧 ∈ [0, ℎ𝑖]} +where 𝐴𝑖 ⊂ 𝑅2 is the object’s footprint and ℎ𝑖 is the height of the object. The set of points along the boundary of 𝐴𝑖 is a +simple two-dimensional polygon denoted 𝜕𝐴𝑖 whose shape is defined by an ordered set of points with a positive signed +area. Points on the interior of 𝐴𝑖 belong to the set denoted int(𝐴𝑖). The polygonal areas of each object do not intersect +int(𝐴𝑖) ∩ int(𝐴 𝑗) = ∅ for all 𝑖 ≠ 𝑗 with 𝑖, 𝑗 ∈ {0, . . . , 𝑁𝑂 − 1}. The height of the tallest object in 𝑂 is denoted ℎmax, +and the airplane is constrained to fly in a feasible airspace +𝐹 = 𝐷 × [𝑧min, 𝑧max] − 𝑂 , +(4) +where 𝐷 ⊂ R2 is the planar region containing the polygonal objects, i.e., 𝐴𝑖 ⊂ 𝐷 for all 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1}, 𝑧min and +𝑧max > 𝑧min are the minimum and maximum operating altitudes of the airplane. The union of all the objects is subtracted +from the rectangular volume 𝐷 × [𝑧min, 𝑧max] in (4). To ensure that 3D Dubins paths joining two configurations does not +exceed the feasible airspace or encounter obstacles, the feasible airspace and set of objects can be artificially contracted +and inflated, respectively. This work assumes that the minimum altitude 𝑧min is constrained to be above the tallest +3 + +building, 𝑧min > ℎmax + 2𝜌min, such that the airplane’s feasible airspace is free of objects and there is enough vertical +space to maneuver without collision. +C. Target Visibility Volumes +The airplane is assumed to be equipped with a gimbaled camera and is tasked with inspecting a set of 𝑀 targets +located at the points 𝑃 = { 𝒑0, . . . , 𝒑𝑀−1}. Each target 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) ∈ 𝑃 is located in an unobstructed area of the +ground plane or on the exposed surface of an object. That is, each target has planar location (𝑝𝑥, 𝑝𝑦) ∈ 𝐷 and altitude +𝑝𝑧 satisfying the following cases: (i) if (𝑝𝑥, 𝑝𝑦) ∩ 𝐴𝑖 = ∅ for all 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1} then the target is on the ground +plane with 𝑝𝑧 = 0, (ii) if 𝑝𝑦 ∩ 𝜕𝐴𝑖 ≠ ∅ for some 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1} then the target is located on the vertical wall of the +𝑖th object and 𝑝𝑧 ∈ [0, ℎ𝑖], or (iii) if 𝑝𝑦 ∩ int(𝐴𝑖) ≠ ∅ then 𝑝𝑧 = ℎ𝑖 such that the target is on top of the 𝑖th object. For +each target, a target visibility volume 𝑉𝑖 is defined as the set of points 𝒈 ∈ R3 that have a direct line-of-sight to the target +(i.e., not obscured by buildings). Let +𝐿(𝜏; 𝒈, 𝒑) = ( 𝒑 − 𝒈)𝜏 + 𝒈 for 𝜏 ∈ [0, 1] +(5) +denote a line segment that joints two points 𝒈, 𝒑 ∈ R3 where 𝜏 is a normalized arc-length. The visibility volume for a +target located at 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) is the subset of the feasible airspace that is within direct line-of-sight to the target, +within a maximum range 𝑑max relative to the target, and at least a distance ℎview above the target: +𝑉( 𝒑; 𝐹, 𝑂, 𝑑max, ℎview) = {𝒈 = (𝑔𝑥, 𝑔𝑦, 𝑔𝑧) ∈ 𝐹 such that || 𝒑 − 𝒈|| ≤ 𝑑max, ℎview + 𝑝𝑧 ≤ 𝑔𝑧 and +𝐿(𝜏; 𝒈, 𝒑) ∩ 𝑂 𝑗 = ∅ for all 𝜏 ∈ [0, 1] and 𝑗 ∈ {0, . . . , 𝑁𝑂 − 1}} . +(6) +For brevity, visibility volumes (6) are henceforth denoted 𝑉( 𝒑). The maximum range 𝑑max constraint models minimum +image resolution requirements. The minimum height-above-target ℎview < 𝑑max constraint ensures images are captured +with sufficient surrounding context (e.g., the point target may actually represent an extended body that should be +contained in the image) or to reduce gimbal pointing speed and precision requirements. For the problem to be well +posed, there should always exist at least one valid viewing point above each target. This condition may be satisfied by +the following parameter constraints: +𝑧min ≤ ℎview + ℎmax ≤ 𝑧max , +(7) +𝑧min ≤ 𝑑max , +(8) +2𝑑max < || 𝒑𝑖 − 𝒑 𝑗|| +for all 𝒑𝑖, 𝒑𝑖 ∈ 𝑃 with 𝒑𝑖 ≠ 𝒑 𝑗 . +(9) +If a target is located on top of the highest object, then constraint (7) ensures that a viewing point exists that is below the +maximum feasible altitude and above the minimum feasible altitude. For targets that are located on the ground plane, +constraint (8) ensures that the sensor range is sufficiently large to view the target from the minimum feasible altitude. +Lastly, constraint (9) is a simplifying assumption that guarantees targets are spaced sufficiently far apart such that their +visibility volumes do not intersect 𝑉( 𝒑𝑖) ∩ 𝑉( 𝒑 𝑗) = ∅ for all 𝑖, 𝑗 ∈ {0, . . . , 𝑀 − 1} with 𝑖 ≠ 𝑗. +D. View-planning Problem Statement +Let 𝐵(𝒒) be a mapping from a configuration 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 to an integer in the set {0, . . . , 𝑀 − 1} that +identifies the visibility volume corresponding to 𝒒, i.e., the integer 𝐵(𝒒) corresponds to the target 𝒑𝐵(𝒒) ∈ 𝑃 for which +(𝑥, 𝑦, 𝑧) ∈ 𝑉( 𝒑𝐵(𝒒)). If 𝒒 is not contained in any visibility volume then 𝐵(𝒒) = ∅. The optimization problem is to find +the sequence of vehicle configurations 𝒒0, . . . , 𝒒𝑀−1 that +minimize +𝑀−1 +∑︁ +𝑖=0 +𝐷(𝒒𝑖, 𝒒𝑖+1) + 𝐷(𝒒𝑀−1, 𝒒0) , +(10) +subject to +𝐵(𝒒𝑖) ≠ 𝐵(𝒒 𝑗), +for all 𝑖, 𝑗 ∈ {0, . . . , 𝑀 − 1} with 𝑖 ≠ 𝑗 , +(11) +𝐵(𝒒0) ∪ · · · ∪ 𝐵(𝒒𝑀−1) = {0, . . . , 𝑀 − 1} , +(12) +where the cost function (10) is the total length of the 3D Dubins paths in the tour, the constraint (11 ensures that each +vehicle configuration lies within a unique visibility volume and the constraint 12) ensures that all visibility volumes +are visited. The view planning problem (10)–(12) is a mixed continuous/combinatorial optimization problem with a +nonlinear cost function and constraints. Since the vehicle travels at a constant speed the minimum-length tour is also the +minimum-time tour. +4 + +III. 2D Algorithms +In this section, a target visibility volume mesh approximation is described (Sec. III.A) followed by a description of +two-dimensional algorithms (Sec. III.B and Sec. III.C) that solve the view planning problem (10)–(12). The algorithms +discussed here include (i) traveling directly over each target (i.e., formulating a Dubins traveling salesperson problem +(DTSP) [12]), and (ii) the DTSP with neighborhoods (DTSPN) to visit one point in a set of visibility polygons +corresponding to the targets [6] that is modified to use an optimized altitude for defining the visibility polygons +A. Target Visibility Volume Approximation +Volumes in 3D are commonly approximated by a triangular mesh [13]. While many prior works on the DTSP +have assumed simplified 3D geometries (e.g., spheres, cylinders), we propose to use triangular meshes since they can +represent arbitrary geometries. The 𝑖th target visibility volume 𝑉𝑖 is approximated with 𝑁𝐹 triangular mesh elements +resulting in the mesh ˆ𝑉𝑖. Let | ˆ𝑉𝑖| denote the total number of mesh elements. The 𝑗th mesh element in ˆ𝑉𝑖 is defined +as a set of vectors ˆ𝑉𝑖 𝑗 = {𝒄0 +𝑖 𝑗, 𝒄1 +𝑖 𝑗, 𝒄2 +𝑖 𝑗, 𝒏𝑖 𝑗} where the vectors 𝒄0 +𝑖 𝑗, 𝒄1 +𝑖 𝑗, 𝒄2 +𝑖 𝑗 ∈ R3 are the positions of the vertices of a +triangular mesh element, and 𝒏𝑖 𝑗 ∈ R3 is an outward pointing normal vector, as illustrated in Fig. 3. The mesh-based +target visibility volumes ˆ𝑉𝑖 are computed using the painter’s algorithm [14]. A sphere centered on each target is +decomposed into six mutually perpendicular views, and each view looks out from the target point location with a +90-degree field-of-view thereby covering one of the six sides of a cube enclosing the point. OpenGL [15] and a special +version of the geometric depth map, i.e., inverse depth, is used to capture the depth of scene objects in the direction of +each view. After calculating the depth values, those that are less than or equal to 𝑑max are tessellated into a preliminary +3D visibility volume mesh. This mesh is genus-0 [13], i.e., a deformation of the sphere, and is also a manifold surface +amenable to constructive solid geometry (CSG) Boolean operations. Next, the mesh is intersected with the feasible +airspace 𝐹 and the minimum viewing distance constraint ℎmin is imposed using CSG Boolean intersection operations. +To reduce the number of vertices in the resulting mesh a decimation procedure is applied [13]. +Fig. 3 +Example target visibility region with a mesh element defined by three vertices 𝒄0, 𝒄1, 𝒄2 and outward +pointing normal vector 𝒏. +B. Baseline Algorithm: Dubins Traveling Salesperson Problem (DTSP) +The DTSP is the problem of finding the shortest planar tour that visits all points in a graph once using points that are +connected with 2D Dubins paths. Since the objects considered here are extruded polygons, there are no features that can +block viewing targets from above (e.g., bridges or tunnels are not admissible). Consequently, the view planning problem +(10)–(12) can be solved with the DTSP by flying at a fixed altitude directly over each target. All feasible altitudes (i.e., +that are common to all visibility volumes) lead to identical cost tours. To account for the different possible heading +angles at each overhead location the heading-angle-discretized DTSP is adopted [16]. An example solution is shown in +Fig. (4a). +5 + +C. Optimized Altitude DTSP with Neighborhoods (DTSPN) +A more sophisticated approach developed by Obermeyer et al. [6] considers the fixed altitude slices of the target +visibility volumes (i.e., planar visibility polygons). Vehicle configurations in each visibility polygon are sampled and a +DTSPN [5, 6] is formulated to visit one configuration in each visibility polygon. In [6], two sampling algorithms were +proposed: entry pose sampling—wherein samples are made along the edge of the polygon with heading angles that are +tangent or inward pointing (Fig. 4b)—and interior pose sampling—wherein samples are placed uniformly in a grid on +the interior of the visibility polygons with uniformly sampled heading angles. In [6], entry pose sampling gave lower +cost solutions than interior pose sampling. Thus, the entry pose sampling method is adopted here. The constraints of +the view planning problem (7)–(9) allow for visibility volumes to occupy disjoint segments of altitude. That is, there +may not exist an altitude 𝑧∗ ∈ [𝑧min, 𝑧max] that is common to all visibility volumes. While this does not pose an issue +for some of the 3D algorithms proposed later, these cases cannot be solved by the 2D (constant-altitude) algorithms +described here. However, introducing the additional constraint +ℎmax + ℎview ≤ 𝑑max +(13) +ensures that the visibility volumes for a target located on the ground plane and for a target located atop the highest +object have at least one common altitude at 𝑧∗ = 𝑑max. In general, there is a range of admissible altitudes 𝑧∗ that may +be chosen. The choice of altitude impacts the 2D DTSPN algorithm since visibility polygons change in shape and +size as the altitude varies. Intuitively, larger polygons are preferred over smaller ones since this increases the set of +candidate configurations. This work proposes to identify an optimal working altitude for the 2D algorithm as follows. +First, 𝑛slice polygons are generated from each visibility volume mesh (i.e., for all targets) using the method described +in [17]. Let P = polygonFromMesh( ˆ𝑉, 𝑧) denote the polygon that results from slicing mesh ˆ𝑉 at altitude 𝑧 and let +polygonArea(P) denote the corresponding area. The optimal altitude 𝑧∗ is chosen as the one that maximizes the sum +of polygonal areas across all visibility polygons: +𝑧∗ = argmax +𝑧∈𝑍 +𝑀−1 +∑︁ +𝑖=0 +polygonArea(polygonFromMesh( ˆ𝑉𝑖, 𝑧)) +(14) +where 𝑍 = {𝑧0, . . . , 𝑧𝑛slice−1} is a set of altitudes at which the polygons are computed. Note that all altitudes 𝑧 ∈ 𝑍 are +constrained such that 𝑧min ≤ 𝑧 ≤ 𝑧max. Also, note that 𝑧0 and 𝑧slice−1 are not 𝑧min and 𝑧max respectively, instead 𝑧0 and +𝑧slice−1 are slightly offset into the body to avoid floating point error. The selection of 𝑧∗ is visualized in Fig 5a. +(a) 2D Dubins traveling salesperson problem (DTSP) +(b) 2D DTSP with neighborhoods (DTSPN) +Fig. 4 +Example solutions to the view-planning problem using 2D constant-altitude algorithms. The green +regions are visibility polygons for a chosen altitude 𝑧∗ while the black arrows represent the heading angle at +sampling points. The multicolored line is the solution path of the TSPs with increasing path length represented +by color changes from red to purple. The DTSP (a) is solved with entry pose sampling using eight headings +samples directly over the targets, while the 2D DTSPN (b) uses eight sample locations around the perimeter of +the polygon with four heading angles that are tangent or point into the corresponding visibility polygons. +IV. 3D Algorithms +Inspection tours that admit three-dimensional maneuvering can potentially lead to path length reductions when +compared to two-dimensional (constant-altitude) tours. The solution techniques in this work all use a transformation +6 + +approach to solve the (2D or 3D) DTSPN according to the following steps: compute the approximation of the +target visibility volume (Sec. III.A), sample the visibility volumes to create graph vertices corresponding to vehicle +configurations, calculate edge costs between vertices using the 3D Dubins path planning algorithm, and solve for a +DTSPN tour. Section IV.A details three algorithms to sample the target visibility volumes: random face sampling, 3D +edge sampling, and global weighted face. Section IV.B.2 then introduces a heuristic approach that improves the edge +cost computation time using a modified Euclidean distance edge cost and a geometric approach to assign heading and +pitch angles at each configuration in the tour. +(a) Optimized altitude with entry pose sampling +(b) Random face sampling +(c) 3D edge sampling +(d) Global weighted face sampling +Fig. 5 +Visualizations of the four different sampling algorithms: optimized altitude with entry pose sampling, +random face sampling, 3D edge sampling, and global weighted face sampling. Two-dimensional representations +of the visibility volumes are in gray, altitude slices are orange lines, and sampled configurations are blue circular +markers. +A. Sampling Strategies +1. Random Face Sampling +The random face sampling algorithm extends the 2D entry pose strategy from [6] to sample 3D vehicle configurations +across the surface of the target visibility volume with a uniform distribution. The approach is detailed in Algorithm 1 +and visualized in Fig. 5b. The algorithm randomly finds 𝑛pts three-dimensional points on the faces of each triangular +mesh in the set of triangular meshes ˆ𝑉0:𝑀−1 = { ˆ𝑉1, . . . , ˆ𝑉𝑀−1} and assigns to each point a set of configurations with +𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively. The sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle +configurations per visibility volume. First, the set of configurations Q is initialized as an empty set, and the area of +each face in the mesh is calculated (lines 3-6). The area of each triangular face element, 𝑎𝑖 𝑗, is calculated by the +elementArea function using +𝑎𝑖 𝑗 = elementArea( ˆ𝑉𝑖 𝑗) = 1 +2 ||(𝒄0 +𝑖 𝑗 − 𝒄1 +𝑖 𝑗) × (𝒄0 +𝑖 𝑗 − 𝒄2 +𝑖 𝑗)|| , +(15) +where × is the vector cross product and 𝒄0 +𝑖 𝑗, 𝒄1 +𝑖 𝑗, and 𝒄2 +𝑖 𝑗 are the three vertices contained in the triangular face element +ˆ𝑉𝑖 𝑗. Next, the proportion of each face area to the total surface area of the mesh ˆ𝑉𝑖 is calculated using element-wise +division (line 7). The randomSetOfIndices function identifies 𝑛pts random faces by sampling faces with probability +in proportion to the weights 𝒘𝐹 (line 8). The use of the proportional surface area during the random selection process +gives every point on the surface of the target visibility region an equal chance of being selected. For each selected +triangular face, a point on the face is randomly selected using a Barycentric coordinate system [18] (lines 10-12). The +Barycentric coordinate system allows for the mapping of two random numbers 𝑟0 and 𝑟1 sampled uniformly from the +interval [0, 1] onto a triangle, embedded in R3, with the weighted sum of its vertices [18]. The random numbers 𝑟0 and +𝑟1 are first sampled (line 11) and then a position on the chosen triangular face is determined (line 12). For each position, +𝑛𝜓 heading angles sampled uniformly between 0 and 2𝜋 as well as 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled +uniformly then added to the set of vehicle configurations. The runtime of the algorithm is dominated by the nested for +loops on lines 2 and 4 running 𝑀| ˆ𝑉|max times—where | ˆ𝑉|max = max𝑖∈{0,1,...,𝑀−1}(| ˆ𝑉𝑖|) is the maximum number of +7 + +faces in a single mesh—and the collections of nested loops on lines 13-14 which run 𝑀𝑛pts𝑛𝜓𝑛𝛾. When the number of +mesh faces in a visibility volume is greater than the number of samples collected | ˆ𝑉|max > 𝑛pts𝑛𝜓𝑛𝛾 the runtime is +𝑂(𝑀| ˆ𝑉|max). +Algorithm 1 Random Face Sampling +function: RandomFaceSampling( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝛾max, 𝛾min) +input: target visibility volume mesh ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of +pitch angles 𝑛𝛾, max pitch angle 𝛾max, min pitch angle 𝛾min +output: a set of vehicle configurations for each target Q +1: Q ← ∅ +2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do +3: +Q𝑖 ← ∅, 𝒂𝑖 ← ∅ +4: +for ˆ𝑉𝑖 𝑗 ∈ ˆ𝑉𝑖 do +5: +𝒂𝑖 ← 𝒂𝑖 ∪ elementArea( ˆ𝑉𝑖 𝑗) +6: +end for +7: +𝒘𝐹 = 𝒂𝑖/�|𝒂𝑖 |−1 +𝑗=0 +𝑎𝑖 𝑗 +8: +𝐼 ← randomSetOfIndices(𝑛pts, 𝒘𝐹) +9: +for 𝑖 ∈ 𝐼 do +10: +𝒄0 +𝑖 𝑗, 𝒄1 +𝑖 𝑗, 𝒄2 +𝑖 𝑗 ← getVertices( ˆ𝑉𝑖 𝑗) +11: +𝑟0 ∼ U[0,1], 𝑟1 ∼ U[0,1] +12: +𝒔 ← 𝒄0 +𝑖 𝑗 (1 − √𝑟0) + 𝒄1 +𝑖 𝑗 +√𝑟0(1 − 𝑟1) + 𝒄2 +𝑖 𝑗 +√𝑟0𝑟1 +13: +for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do +14: +for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do +15: +Q𝑖 ← Q𝑖 ∪ �𝒔, 2𝑗𝜋/𝑛𝜓, 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1)� +16: +end for +17: +end for +18: +end for +19: +Q ← Q ∪ Q𝑖 +20: end for +2. 3D Edge Sampling +The second sampling strategy proposed is 3D edge sampling wherein the 2D entry pose strategy from [6] is extended +to sample 3D vehicle configurations across the lowest feasible altitude. For the visibility volume shapes studied here this +is also the altitude where the cross-sectional area is largest for each shape. The 3D edge sampling algorithm, detailed +in Algorithm 2 and visualized in Fig. 5c, finds 𝑛pts three-dimensional points on the polygon created by slicing the +triangular mesh along the lowest feasible altitude and distributing points uniformly along the perimeters. The algorithm +then assigns a set of configurations to each point with 𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively. The +sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle configurations per visibility volume. First, the set of configurations +Q is initialized as an empty set (line 1). Then, for each visibility volume a subset of points contained in that volume +is initialized (line 3). Next, the 𝑧 minimum altitude for the triangular mesh is found by finding the minimum height +coordinate in the set of vertices in the mesh ˆ𝑉 𝑧 +𝑖 (line 4). After, the polygonFromMesh algorithm takes the triangular +mesh and the 𝑧min altitude and returns a polygonal slice of the mesh (line 5). A set of points, 𝝀 ∈ R2, placed uniformly +along the edge of the polygon is found using the uniformPerimeterPoints which takes a polygon and the number +of points desired as arguments (line 6). Note that lines 4–6 can be modified to produce samples at multiple altitude +slices if desired. Next, the algorithm iterates through each sampled point and assigns heading and pitch angles. To +ensure inward-pointing heading angles, the direction of the line segment containing the sample point is found using the +tangentAngle function (line 8). The points in the polygon defined by polygonFromMesh have a positive signed area. +Thus, the inward-pointing heading angles are the angles from [0, 𝜋] measured counter-clockwise from the tangent angle. +For each position, 𝑛𝜓 heading angles between 𝜓𝑞 and 𝜓𝑞 + 𝜋 and 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled +uniformly and returned as part of the vehicle configurations (lines 9-15). To achieve 𝑛 equally spaced angle samples, +including the minimum and maximum angle, the range is divided into 𝑛 − 1 sub-sections. The max function, on lines 10 +and 12, ensures the range is never divided by zero (the case where 𝑛𝛾 or 𝑛𝜓 is one). The runtime complexity is dominated +8 + +by the for loops on lines 2-18 which have a worst-case running time complexity of 𝑂(𝑀𝑘), where 𝑘 = | ˆ𝑉|2 +max + 𝑛pts𝑛𝜓𝑛𝛾. +| ˆ𝑉|2 +max is the runtime of the polygonFromMesh algorithm while 𝑛pts𝑛𝜓𝑛𝛾 is the runtimes for the nested for loops (lines +9-15). For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater +than the total number of configurations returned, | ˆ𝑉|2 +max > 𝑛pts𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀| ˆ𝑉|2 +max). +Algorithm 2 3D Edge Sampling +function: 3DEdgeSampling( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝛾max, 𝛾min) +input: target visibility volume mesh ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of +pitch angles 𝑛𝛾, max pitch angle 𝛾max, min pitch angle 𝛾min +output: a set of vehicle configurations for each target Q +1: Q ← ∅ +2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do +3: +Q𝑖 ← ∅ +4: +𝑧min ← min( ˆ𝑉 𝑧 +𝑖 ) +5: +P ← polygonFromMesh(𝑧min, ˆ𝑉𝑖) +6: +{𝝀0, . . . , 𝝀𝑛pts−1} ← uniformPerimeterPoints(P, 𝑛pts) +7: +for 𝑚 ∈ {0, . . . , 𝑛pts − 1} do +8: +𝜓𝑞 ← tangentAngle(𝝀𝑚, P) +9: +for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do +10: +𝜓 ← 𝜓𝑞 + 𝑗𝜋/max(𝑛𝜓 − 1, 1) +11: +for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do +12: +𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1) +13: +Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧min, 𝜓, 𝛾) +14: +end for +15: +end for +16: +end for +17: +Q ← Q𝑖 ∪ Q +18: end for +3. Global Weighted Face Sampling +The third proposed sampling strategy is global weighted face sampling. Rather than sampling the visibility volumes +at the lowest altitude, all target visibility volumes are sampled along a common set of altitude planes and the number of +samples allocated to each plane is determined by the cross-sectional perimeter distribution of each altitude summed +across all target visibility volumes. This approach places more samples at altitudes common to all targets that, on +average, also have large cross-sectional areas. This sampling method is detailed in Algorithm 3 and visualized in Fig. 5d. +The algorithm takes a set of target visibility meshes ˆ𝑉0:𝑀−1 and returns a set of vehicle configurations Q for each mesh +given the parameters 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝑛slice, 𝛾max, and 𝛾min where 𝑛slice ≥ 2 is the number of altitude slices to consider. Let +ˆ𝑉 𝑧 +0:𝑀−1 denote the set of all 𝑧 heights for every vertex contained across the 𝑀 meshes ˆ𝑉0:𝑀−1. First, the global minimum, +the global maximum altitude, and the slicing altitude step size are found (lines 1-2). Then a vector 𝝁 is initialized +with zeros, denoted as 0𝑛slice×1 (line 3), and later stores the total perimeter summed across all visibility polygons at the +corresponding altitude slice. The target visibility volumes are sliced into polygons with fixed altitude (i.e. parallel to +the 𝑥𝑦 plane) using the polygonFromMesh function, lines 4-9. The lowest 𝑧 plane is the visibility volumes’ global +minimum 𝑧 height (𝜁min) and the highest 𝑧 plane is the visibility volumes’ global maximum 𝑧 height (𝜁max), line 1. The +nominal set of altitude planes is then 𝑍 = {𝑧0, . . . , 𝑧𝑛slice−1} where 𝑧0 = 𝜁min, 𝑧𝑛slice−1 = 𝜁max and 𝑧𝑖+1 − 𝑧𝑖 = 𝐿. At each +plane 𝑧 ∈ 𝑍, polygons are created from the target visibility volume and the polygons’ perimeters are accumulated, line 7. +The sample points in each 𝑧 plane are then distributed in proportion to the accumulated perimeters, lines 11-29. The +function iteratePerimeters takes six arguments: the mesh to iterate across, a perimeter distribution, the minimum +altitude, the maximum altitude, the step size, and the total number of sample points. It returns a variable number of +𝑛𝑧 ≤ 𝑛slice elements where each element is a pair consisting of a 𝑧𝑟 altitude and the number of points to sample at that +altitude, 𝑛𝑟. An altitude slice 𝑧𝑟 is either an element of 𝑍 and/or an altitude located at the top or bottom of each visibility +volume. At each altitude 𝑧𝑟 the corresponding value of 𝝁 is determined (or interpolated, in the special case that 𝑧𝑟 ∉ 𝑍) +and the 𝑛pts are distributed to each 𝑧𝑟 in proportion to the result. In the event that no slices intersect the visibility mesh +9 + +then 𝑛𝑧 = 2 and the heights 𝑧𝑟 are returned corresponding to the top and bottom of the target visibility volume. Next, +samples 𝝀 ∈ {𝝀0, ..., 𝝀𝑛𝑟−1} are placed uniformly around the perimeter of each polygon created by the intersection of +the 𝑧𝑟 planes and the target visibility volume with the function uniformPerimeterPoints, line 16. The heading and +pitch angles are sampled in the same way as entry pose sampling [6], pointing tangent or inward with respect to the +polygon. The angle tangent to each point 𝝀 on the perimeter of the polygon is found with the tangentAngle function. +The pitch angles are uniformly sampled within the pitch angle constraints. The runtime complexity is dominated by the +for loops on lines 11-29 which have a worst-case running time complexity of 𝑂(𝑀𝑛slice𝑘), where 𝑘 = | ˆ𝑉|2 +max + 𝑛𝑟𝑛𝛾𝑛𝜓. +For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater than the +total number of configurations returned, | ˆ𝑉|2 +max > 𝑛𝑟𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀𝑛slice| ˆ𝑉|2 +max). +Algorithm 3 Global Weighted Face +function: GlobalWeightedFace( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝑛slice, 𝛾max, 𝛾min) +input: set of triangular meshes ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of pitch +angles 𝑛𝛾, number of altitude slices 𝑛slice, max pitch angle 𝛾max, min pitch angle 𝛾min +output: a set of vehicle configurations for each target Q +1: 𝜁max ← max( ˆ𝑉 𝑧 +0:𝑀−1), 𝜁min ← min( ˆ𝑉 𝑧 +0:𝑀−1) +2: 𝐿 ← (𝜁max − 𝜁min)/(𝑛slice − 1) +3: 𝝁 ← 0𝑛slice×1 +4: for 𝑖 ∈ {0, . . . , 𝑛slice − 1} do +5: +for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do +6: +P ← polygonFromMesh(𝜁min + 𝐿𝑖, ˆ𝑉𝑖) +7: +𝜇𝑖 ← 𝜇𝑖 + perimeter(P) +8: +end for +9: end for +10: Q ← ∅ +11: for ˆ𝑉𝑖 ∈ ˆ𝑉1:𝑀 do +12: +(𝑧𝑟, 𝑛𝑟)𝑛𝑧−1 +𝑟=0 +← iteratePerimeters( ˆ𝑉𝑖, 𝝁, 𝜁min, 𝜁max, 𝐿, 𝑛pts) +13: +Q𝑖 ← ∅ +14: +for 𝑟 ∈ {0, . . . , 𝑛𝑧 − 1} do +15: +P ← polygonFromMesh(𝑧𝑟, ˆ𝑉𝑖) +16: +{𝝀0, . . . , 𝝀𝑛𝑟−1} ← uniformPerimeterPoints(P, 𝑛𝑟) +17: +for 𝑚 ∈ {0, . . . , 𝑛𝑟 − 1} do +18: +𝜓𝑞 ← tangentAngle(𝝀𝑚, ˆ𝑉𝑖) +19: +for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do +20: +for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do +21: +𝜓 ← 𝜓𝑞 + 𝑘𝜋/max(𝑛𝜓 − 1, 1) +22: +𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1) +23: +Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧𝑟, 𝜓, 𝛾) +24: +end for +25: +end for +26: +end for +27: +end for +28: +Q ← Q𝑖 ∪ Q +29: end for +B. Proposed Heuristics +1. Modified Euclidean Distance Traveling Salesperson Problem with Neighborhoods (METSPN) +A bottleneck in the 3D DTSPN algorithms is the computation of the edge costs that require solving for a 3D +Dubins path between two configurations 𝒒𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖, 𝜓𝑖, 𝛾𝑖) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗). Since the Dubins path is +asymmetric the corresponding edge cost must be computed for each direction. Here, we propose an approximation to +10 + +this edge cost +ˆ𝐷(𝒒𝑖, 𝒒 𝑗) = max +� +|𝛿𝑧| +sin 𝛾limit +, ∥𝒔𝑖 − 𝒔 𝑗 ∥2 +� +, +(16) +where 𝛿𝑧 = 𝑧 𝑗 − 𝑧𝑖, 𝛾limit = 𝛾max if 𝛿𝑧 > 0 and 𝛾limit = 𝛾min otherwise, 𝒔𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and 𝒔 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗). The +calculation is visualized in Fig. 6. The distance (16) is a lower bound on the actual 3D Dubins path length, i.e., +Fig. 6 +Visualization of the modified Euclidean distance. The Euclidean distance shown in blue with a pitch +angle 𝛾 > 𝛾limit is modified by extending the distance traveled in the 𝑥𝑦 plane resulting in the red line with pitch +angle 𝛾limit. ˆ𝐷𝑥𝑦 refers to the length of the Dubins path projected onto the 𝑥𝑦 plane. +ˆ𝐷(𝒒0, 𝒒1) ≤ 𝐷(𝒒0, 𝒒1), and is significantly faster to compute than solving for the Dubins path. Using this edge cost +leads to a variant of the DTSPN we refer to as the modified Euclidean distance traveling salesperson problem (METSPN). +Solving the METSPN gives a tour of 3D locations to visit. Once a tour is found for the METSPN it is converted into a +feasible sequence of Dubins paths by assigning heading and pitch angles as follows. +2. Bisecting Angle Approximation +To assign heading and pitch angles a heuristic is adopted that extends the mean angle algorithm developed in [19] to +three dimensions. The approach is summarized in Algorithm 4. The proposed bisecting angle approximation takes +as parameters: 𝑽 a 𝑀 × 3 matrix corresponding to the sequence of vertices in the METSPN tour and the problem +parameters: 𝜌min, 𝛾min, and 𝛾max. The algorithm returns a set of vehicle configurations Q at each point in 𝑽 with +heading and pitch angles defined as the angle bisector of each consecutive triplet of vertices (for points spaced far apart) +or as a straight segment (for points spaced close together). +To obtain the angle bisector at each vertex, calculate vectors from the preceding vertex 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 = (𝑢𝑥, 𝑢𝑦, 𝑢𝑧) +and to the following vertex 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 = (𝑤𝑥, 𝑤𝑦, 𝑤𝑧) (line 3). The vector 𝒃 = 𝒘 + 𝒖 = (𝑏𝑥, 𝑏𝑦, 𝑏𝑧) determines the +heading angle 𝜓 in the 𝑥𝑦 plane computed with the four-quadrant arctangent function (line 4). A visualization of the +calculation can be seen in Fig. 7. The circular indexing of 𝑽, a 𝑀 by 3 matrix, allows for the index −1 to refer to the last +Fig. 7 +The notation used to determine the bisector vector for a triplet of three points: 𝑽𝑖−1, 𝑽𝑖, 𝑽𝑖+1. The +orientation of the vectors 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 and 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 are summed and normalized resulting in the vector +𝒗. The heading angle 𝜓 is the component of 𝒃 in the 𝑥𝑦 plane while the pitch angle 𝛾 is measured from the 𝑥𝑦 +plane. +column of 𝑽 and the index 𝑛 to refer to the first element of 𝑽. The pitch angle bisector is the angle between the vector 𝒃 +11 + +and the 𝑥𝑦 plane (line 5). The resulting angle is saturated to be within the pitch angle bounds on line 6. If vertices are +close together then curve-curve-curve (CCC) Dubins paths may be created. This should be avoided because the cost of +(CCC) Dubins paths is much greater than the Euclidean distance. The likelihood of CCC paths occurring is reduced by +setting the heading and pitch in the direction of the line between two vertices. If the distance between two vertices is +small (less than the long path case in [20]), then heading and pitch angles are aligned with the while loop on lines 10-21. +To align the headings of two configurations, the vector between the internal coordinates is found. The angle of this +vector, 𝒘, about the 𝑧 axis is used as the heading angle. Then, the angle between the 𝑥𝑦 plane and the vector 𝒘 is found +and saturated between 𝛾min and 𝛾max to set the pitch angle. Inside the loop, the index is advanced once but it is also +advanced a second time if the current vertex and the next vertex are within 4𝜌min units of each other (worst case for +the long path case [20]). The second index advance is required to pass over the next configuration because it was just +modified. +Algorithm 4 Bisect Angle Approximation +function: BisectAngleApprox(𝑽, 𝜌min, 𝛾min, 𝛾max) +input: 𝑽 is a 𝑛 by 3 matrix of vertices that solve the METSPN, minimum turn radius 𝜌min, minimum pitch angle 𝛾min, +maximum pitch angle 𝛾max +output: set of configurations solving a DTSP Q +1: Q ← ∅ +2: for 𝑖 ∈ {0, 1, 2 . . . 𝑀 − 1} do +3: +𝒃 ← 𝑽𝑖+1 + 𝑽𝑖−1// indexing into 𝑽 is circular +4: +𝜓 ← atan2(𝑏𝑥, 𝑏𝑦) +5: +𝛾 ← atan2(𝑏𝑧, +√︃ +𝑏2𝑥 + 𝑏2𝑦) +6: +𝛾 ← max(min(𝛾, 𝛾max), 𝛾min) +7: +Q ← Q ∪ (𝑽𝑖, 𝜓, 𝛾) +8: end for +9: 𝑖 ← 0 +10: while 𝑖 < |𝑽| do +11: +if ||𝑽𝑖 − 𝑽𝑖+1|| < 4𝜌min then +12: +𝒘 ← 𝑽𝑖+1 − 𝑽𝑖 +13: +𝜓 ← atan2(𝑢𝑥, 𝑢𝑦,) +14: +𝛾 ← atan2(𝑢𝑧, +√︃ +𝑢2𝑥 + 𝑢2𝑦) +15: +𝛾 ← max(min(𝛾, 𝛾max), 𝛾min) +16: +Q𝑖𝜓 ← 𝜓, Q(𝑖+1) 𝜓 ← 𝜓 +17: +Q𝑖𝛾 ← 𝛾, Q(𝑖+1)𝛾 ← 𝛾 +18: +𝑖 ← 𝑖 + 1 +19: +end if +20: +𝑖 ← 𝑖 + 1 +21: end while +C. Illustrative Examples +An example of a view planning solution for five targets scattered around a city model of Charlotte, North Carolina +is shown in Fig. 8a. The example was constructed assuming a Dubins airplane model having a curvature radius of +𝜌min = 40 m and pitch angle constraints 𝛾 ∈ [−𝜋/12, 𝜋/9]. The random-face algorithm was used with 𝑛pts = 8 samples +per visibility volume, 𝑛𝜓 = 4 heading angles per sample, and 𝑛𝛾 = 1 pitch angle per sample-heading angle pair. The +visibility volumes for targets that had no occlusions had a common dome shape, whereas targets located closer to objects +had more arbitrary shapes. Another example Fig. 8b illustrates the solution for five targets in a model of New York City, +New York. This example compares the three-dimensional random-face algorithm with 𝑛pts = 32, 𝑛𝜓 = 8, and 𝑛𝛾 = 3 +pitch angles, to the two-dimensional optimized altitude entry pose sampling algorithm with 𝑛pts = 32, 𝑛𝜓 = 8. The +3D path can change altitude which allowed the algorithm to find a lower cost path of 3920m while the 2D algorithm +maintained constant altitude and found a path of cost 4285m, a 10.9% reduction in path cost. +12 + +(a) 3D DTSP with neighborhoods (DTSPN) in Charlotte, +North Carolina +(b) 3D DTSP with neighborhoods (DTSPN) in New York +City, New York +Fig. 8 +Solutions to the 3D Dubins traveling salesperson problem with neighborhoods. Panel (a) was computed +using the random face algorithm in light blue with 8 samples per target visibility volume, four heading angles +per sample, and one pitch angle per sample-heading angle pair. Panel (b) was computed using the random face +sampling algorithm in dark blue with 𝑛pts = 32 samples per target visibility volume, 𝑛𝜓 = 8 heading angles per +sample, and 𝑛𝛾 = 3 pitch angles per sample-heading pair; the two-dimensional entry pose sampling from [6] in +magenta with 𝑛pts = 32 samples per target visibility volume and 𝑛𝜓 = 4 heading angles per sample. The target +visibility volume is translucent white with black edges and the targets are red spheres. The green spheres are +the vehicle configurations for the solution to the DTSPN. The environment shown is a section of New York City, +New York obtained from the OpenStreetMap database. Building heights are indicated by the varying color +scale from yellow to purple. +V. Numerical Performance Study +The 2D algorithms from Sec. III were compared to the 3D algorithms from Sec. IV through a Monte-Carlo +experiment that randomized target locations and a number of targets located in an urban environment. This section +describes the implementation of the algorithms, the design of the Monte-Carlo study, and discusses the results. +A. Implementation +The algorithms in this work were written in python 3.9 ∗[21] using a number of packages, including Shapely [22] +for polygonal operations and NumPy [23] for working with matrices. The GLKH traveling salesperson solver [24] was +used to solve the generalized traveling salesperson problems that arise from DTSPs. The target visibility volumes were +created with data from OpenStreetMap [25], inverse depth calculations from the target location using OpenGL [15], and +Blender [26] was used for intersecting the triangular meshes within the feasible airspace 𝐹 as well as decimating the +meshes (i.e., reducing the number of triangular faces). This work uses [27] to compute 2D Dubins paths for the 2D +algorithms. The algorithm simulations were performed on an AMD Threadripper 3990X running Ubuntu 20.04 with +one thread allocated to the algorithm. +B. Monte-Carlo Experiment +A Monte-Carlo experiment was designed using the environments described in Table 1. The environments were +created by capturing all of the buildings in a rectangular area in New York City with the OpenStreetMap database and +limiting the building heights to 300 m. Target locations were randomized for each trial and determined by sampling +the environment and placing targets on the ground, the wall of buildings, or the roofs of buildings according to a +user-defined distribution. The radius of the target visibility volumes was 300 m with each target being at least 600 +m apart. The proposed sampling methods and heuristics are independent and studied here in different combinations. +The algorithms parameters were varied as follows: the number of samples per visibility volume was varied between +𝑛pts = {2, 4, 8, 16, 32}, the number of heading angles per sample was 𝑛𝜓 = {2, 4, 8}. To reduce the number of trials, +only one pitch angle (𝑛𝛾 = 1) of 0◦ was used by passing 0◦ for 𝛾min and 𝛾max to the random face sampling (Sec. IV.A.1), +3D edge sampling (Sec. IV.A.2), and global weighted face sampling (Sec. IV.A.3) algorithms. The Dubins airplane had +∗The implementation of this study can be found at https://github.com/robotics-uncc/VisualTour3DDubins. +13 + +Table 1 +Description of environments obtained from an OpenStreetMap database for New York City, USA, and +used for the Monte-Carlo experiment. +Number of Targets +Number of objects +Width +Depth +5 +5624 +1986 m +2090 m +10 +9202 +2809 m +2857 m +15 +11584 +3440 m +3621 m +20 +12119 +3972 m +4181 m +a minimum curvature radius of 𝜌min = 40 m and a pitch angle constrained between -𝜋/12 and 𝜋/9, similar to [10]. A +total of 80 configurations of targets were generated, divided evenly among groups of 5, 10, 15, and 20 targets. Every +combination of algorithm parameters was evaluated with the 80 configurations. The normalized tour cost (total length +of the tour divided by the turn radius) and the computation time were recorded. The algorithms are denoted by acronyms +wherein the prefix is either 2D-DTSP, 2D-DTSPN, 3D-DTSPN, or 3D-METSPN corresponding to the algorithms of +Sections III.B, III.C, IV.A, and IV.B.1, respectively. The 2D-DTSP is followed by a dash and an integer representing the +number of heading angles. The remaining two algorithms are described by a sampling method acronym: entry pose +sampling (ETRY) from Sec. III.C, random face sampling (RFAC) from Sec. IV.A.1, 3D edge sampling (E3D) from +Sec. IV.A.2, or global weighted face sampling (GWF) from Sec. IV.A.3 followed by a dash and an integer representing +the number of heading angles and another dash and an integer representing the number of samples per target visibility +volume (i.e., 2D-DTSPN-ETRY-4-16 corresponds to a 2D DSTPN using entry pose sampling with 4 heading angles and +16 sample points per target visibility region). +1. Analysis of Monte-Carlo Study +In general, two-dimensional methods at a fixed altitude performed better if the targets are all located at similar +heights; whereas, 3D methods trended towards better tour cost when targets occupy a wide range of altitudes. The +median path length, normalized by dividing the cost by the minimum curvature radius, of each view-planning tour (i.e., +cost) of the Monte-Carlo runs for an increasing number of targets, the number of heading angles is held at 𝑛𝜓 = 8, and +the number of pitch angles is held at 𝑛𝛾 = 1 is plotted in Fig. 9. The DTSP algorithms that only visit a single point +(gray) have one location sample per visibility volume but the lines were extended along the abscissa for comparison. +The DTSP is inefficient in our problem because shorter paths can be obtained between targets by flying through the +boundary of their corresponding visibility volumes rather than requiring the paths to pass through the visibility volume +centers. As the number of heading angles increases the mean cost of the solution decreases, as expected. The METSPN +algorithms have a similar cost to the eight sampled heading angle solutions. Most of the medians for different algorithms +approach an asymptote, suggesting that they are converging towards a fixed median tour cost (i.e., further increasing +the number of samples has diminishing returns). For a large number of samples, the proposed random face sampling +algorithm yields a lower tour cost than the optimized altitude 2D algorithm. However, the median of the 3D edge +sampling algorithm is less than the optimal altitude 2D algorithm for all numbers of samples greater than 2. This may +be due to the 3D algorithms spreading their samples across another dimension (altitude). The 3D algorithms that spread +the samples along the vertical dimension of each visibility volume perform worse than the algorithm that only samples +one altitude slice. This suggests that distributing the points in the horizontal plane is more important than distributing +them in the vertical direction for this particular environment and visibility volume. The sensor model creates visibility +volumes with the most horizontal variation at the bottom of the shape as seen in Fig. 8; therefore, sampling the visibility +volumes at the bottom is the best way to produce samples with the greatest horizontal variation. +To isolate the effects of the different sampling methods, the results are examined for the case where the number +of samples is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8, and the number of pitch angles is +𝑛𝛾 = 1. Box plots of those trials can be seen in Fig. 10. The medians of the 3D methods (black bar in the middle +of the colored box) are lower than the medians of the 2D methods suggesting that the 3D methods are able to more +consistently find lower-cost solutions. The difference between medians of 2D and 3D methods grows as the number of +target visibility volumes increases. The range of solutions for the different methods, denoted by the vertical black bars, +is large and suggests that the difference between the solutions produced by the 2D and 3D cases is variable and sensitive +to the environment. The time for each algorithm to execute on a single thread is shown in Fig. 10. It can be seen that +14 + +2 4 +8 +16 +32 +Samples per Target +170 +180 +190 +200 +Tour Cost (nondim.) +5 Targets +2 4 +8 +16 +32 +Samples per Target +225 +250 +275 +Tour Cost (nondim.) +10 Targets +2 4 +8 +16 +32 +Samples per Target +270 +300 +330 +360 +Tour Cost (nondim.) +15 Targets +2 4 +8 +16 +32 +Samples per Target +325 +350 +375 +400 +425 +450 +475 +Tour Cost (nondim.) +20 Targets +Algorithm +2D-DTSP +2D-DTSPN-ETRY +3D-DTSPN-E3D +3D-DTSPN-GWF +3D-DTSPN-RFAC +3D-METSPN-E3D +3D-METSPN-GWF +3D-METSPN-RFAC +Fig. 9 +The line plots show the median non-dimensional tour cost of the different algorithms as the number of +samples per target visibility volume increases. +the algorithms that only consider one point per region have lower execution times than the algorithms that consider +neighborhoods. The 2D ETRY method has a similar execution time to the 3D DTSPN methods. However, the heuristic +METSPN algorithm has a lower execution time compared to the other 3D methods because the graph that it creates +is smaller and less computationally expensive. The results suggest that for a large number of samples the METSPN +algorithm outperforms the 3D DTSPN algorithms since it produces tours of similar cost but with a computation time +that is approximately two orders of magnitude lower. +VI. Conclusion +This paper studied the view planning problem of using a 3D Dubins airplane model to inspect points of interest in +an urban environment in minimum time. Triangular meshes were used to compute approximate visibility volumes that +correspond to locations where an unobstructed view of the target can be obtained while satisfying imaging and altitude +constraints. The mesh-based approach for computing visibility volumes is flexible and can represent more complex +geometries than have previously been considered. A range-based sensor model was assumed here, however mesh-based +view planning can potentially support other sensor models, sensing modalities, and encode sensing performance +15 + +100 +150 +200 +250 +Tour Cost (nondim.) +5 Targets +250 +300 +350 +Tour Cost (nondim.) +10 Targets +200 +225 +250 +Tour Cost (nondim.) +15 Targets +325 +350 +375 +400 +Tour Cost (nondim.) +20 Targets +5 +10 +15 +20 +Number of Targets +100 +101 +102 +103 +104 +Time (s) +Algorithm +2D-DTSP +2D-DTSPN-ETRY +3D-DTSPN-E3D +3D-DTSPN-GWF +3D-DTSPN-RFAC +3D-METSPN-E3D +3D-METSPN-GWF +3D-METSPN-RFAC +Fig. 10 +The box plots show the range of cost across all sets of target visibility volumes when the number of +samples per target visibility is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8 and the number +of pitch angles is held at 𝑛𝛾 = 1. The vertical black bars show the upper and lower quartiles of the data while +the colored sections show the middle quartiles. The black bar in the middle of the box plots is the median of +the data set. The black diamonds are outliers. The line graph shows the increase in computation time as the +number of target visibility volumes increases on a log10 scale. The shaded region around each line shows the +range of computation time. +characteristics. The 3D Dubins airplane model used in this work can, in some circumstances, produce more efficient +inspection tours by exploiting altitude changes that are otherwise not possible with constant-altitude Dubins path tours. +In cases where visibility volumes occupy disjoint altitude segments, the 3D algorithms provide a feasible solution where +the 2D algorithms are not feasible. However, the pitch angle constraints of a Dubins airplane limit the change in altitude +over a tour. Altitude changes are accompanied by an increase in path length and thus are only efficient when they greatly +improve access to the visibility volume. +This work introduced a heuristic that computes edge costs by replacing the 3D Dubins path computation with a +simpler lower bound and assigning heading and pitch angles based on the geometric relation of successive points in a +tour. This strategy provides a similar tour cost to other 3D algorithms that use the exact 3D Dubins path planner for +edge cost computation but with computation time reduced by two orders of magnitude. Future work may consider the +view planning problem in the presence of obstacles that must be avoided, with target visibility volumes that overlap, +and/or with uncertain moving targets to be inspected. +Acknowledgments +This work was supported by the William States Lee College of Engineering at the University of North Carolina at +Charlotte through the Multidisciplinary Team Initiation (MTI) Grant. +16 + +References +[1] Chitsaz, H., and LaValle, S. M., “Time-optimal paths for a Dubins airplane,” 46th IEEE Conference on Decision and Control, +IEEE, 2007, pp. 2379–2384. https://doi.org/10.1109/CDC.2007.4434966. +[2] Ambrosino, G., Ariola, M., Ciniglio, U., Corraro, F., De Lellis, E., and Pironti, A., “Path generation and tracking in 3-D for UAVs,” +Transactions on Control Systems Technology, Vol. 17, No. 4, 2009, pp. 980–988. https://doi.org/10.1109/TCST.2009.2014359. +[3] Ny, J. 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L., “On the construction of an optimal feedback control law for the shortest +path problem for the Dubins car-like robot,” 30th Southeastern Symposium on Systems Theory, 1998, pp. 280–284. +https://doi.org/10.1109/SSST.1998.660075. +18 + diff --git a/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/load_file.txt b/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9af07b4a67ebf896b747e88acfc644a400ec8d45 --- /dev/null +++ b/-dE4T4oBgHgl3EQf3w3V/content/tmp_files/load_file.txt @@ -0,0 +1,849 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf,len=848 +page_content='Planning Visual Inspection Tours for a 3D Dubins Airplane Model in an Urban Environment Collin Hague ∗, Andrew Willis †, Dipankar Maity ‡, Artur Wolek § University of North Carolina at Charlotte, Charlotte, North Carolina, 28223 This paper investigates the problem of planning a minimum-length tour for a three- dimensional Dubins airplane model to visually inspect a series of targets located on the ground or exterior surface of objects in an urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Objects are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='5D extruded polygons representing buildings or other structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A visibility volume defines the set of admissible (occlusion-free) viewing locations for each target that satisfy feasible airspace and imaging con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The Dubins traveling salesperson problem with neighborhoods (DTSPN) is extended to three dimensions with visibility volumes that are approximated by triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Four sampling algorithms are proposed for sampling vehicle configurations within each visibility volume to define vertices of the underlying DTSPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Additionally, a heuristic approach is pro- posed to improve computation time by approximating edge costs of the 3D Dubins airplane with a lower bound that is used to solve for a sequence of viewing locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The viewing locations are then assigned pitch and heading angles based on their relative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The proposed sampling methods and heuristics are compared through a Monte-Carlo experiment that simulates view planning tours over a realistic urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Introduction U nmanned aerial vehicles (UAVs) are routinely used in applications such as visual reconnaissance, infrastructure inspection, and aerial photography to image a series of points of interest (henceforth referred to as targets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In three-dimensional environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', an urban city, mountainous terrain) the targets must be imaged from particular vantage points to avoid occlusions from surrounding objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', buildings, trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Additional requirements, such as airspace restrictions and image resolution, further constrain the three-dimensional visibility volume from which an image of a target may be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This paper investigates the problem of planning a path to image a set of targets by flying through their corresponding visibility volumes in minimum time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The UAV is modeled as a Dubins airplane [1, 2] and the environment consists of extruded polygonal objects with targets located on the ground or on the surface of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Relation to Prior Work The view planning problem considered here is related to the Dubins traveling salesperson problem (DTSP [3]) of constructing a minimum-time tour for a constant-speed planar Dubins vehicle model [4] to travel through a series of planar points (with arbitrary heading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The set of points to visit can be generalized to arbitrary planar regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', polygons) to give the DTSP with neighborhoods (DTSPN [5]) wherein the Dubins vehicle must visit at least one point in each region/neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' One application of the DTSPN is to plan visual inspection tours for an airplane to visit planar polygonal regions at a constant altitude to image ground targets [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' More recently, the Dubins airplane model [1, 2] that includes additional degrees of freedom (altitude and pitch angle) was used to extend the DTSPN to three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Planning three-dimensional Dubins tours have typically assumed that the desired viewing regions have relatively simple geometries, such as spheres [7] or cylinders [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In contrast, this work admits more complex target visibility volumes that are approximated as triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Contributions This paper formulates a view planning problem for a 3D Dubins airplane model to observe a set of targets occluded by objects in an urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The contributions of the paper are: (1) four sampling algorithms that extend ∗Graduate student, Department of Mechanical Engineering and Engineering Science †Associate Professor, Department of Electrical and Computer Engineering ‡Assistant Professor, Department of Electrical and Computer Engineering §Assistant Professor, Department of Mechanical Engineering and Engineering Science, Member AIAA 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='05309v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='SY] 12 Jan 2023 two-dimensional Dubins-based view planning to three dimensions with visibility volumes that have an arbitrary geometry approximated by a triangular mesh, and (2) a heuristic approach that solves for a tour using a modified Euclidean distance TSP (METSP) with edge costs that are lower bounds for the 3D Dubins path length and using the geometry of consecutive viewing locations in the METSP tour to assign heading and pitch angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The relative performance of the algorithms are characterized through a Monte-Carlo experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Paper Organization The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section II describes the airplane motion model, the environment model, the target visibility volumes, and states the view planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section III describes a method for approximately computing the target visibility volumes and path planning for constant-altitude 2D tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section IV introduces 3D path planning algorithms and proposes heuristics to reduce computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section V describes the results of a Monte-Carlo experiment that compares the 2D and 3D algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The paper is concluded in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Problem Formulation This section formulates the problem of planning a minimum time path for an unmanned airplane to visually inspect a set of targets in the presence of occluding structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The vehicle motion model, environmental model, and target visibility volumes are introduced, and the view planning problem is formally stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Airplane Motion Model This work considers the three-dimensional Dubins airplane model [9, 10]: ������������ �𝑥 �𝑦 �𝑧 �𝜓 �𝛾 ������������ = ������������ 𝑣 cos 𝜓 cos 𝛾 𝑣 sin 𝜓 cos 𝛾 𝑣 sin 𝛾 𝑢𝜓 𝑢𝛾 ������������ , (1) where (𝑥, 𝑦, 𝑧) ∈ R3 is the inertial position of the airplane expressed in an east-north-up coordinate system, 𝑣 is the vehicle’s speed, 𝜓 is the heading angle, and 𝛾 is the pitch angle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The control inputs are the turn-rate 𝑢𝜓 and the pitch-angle-rate 𝑢𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The Dubins airplane model travels in the direction it is pointed so that the pitch angle 𝛾 is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 1 The model for a Dubins airplane flying at speed 𝑣 where (𝑥, 𝑦, 𝑧) is the inertial position, 𝜓 is the heading angle, and 𝛾 is the pitch angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' equivalent to the flight path angle and is constrained between a minimum and maximum angle, 𝛾 ∈ [𝛾min, 𝛾max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The controls are constrained such that the path curvature 𝜌min is bounded [11]: 𝜌min ≤ 1 √︃ 𝑢2 𝜓 cos2 𝛾 + 𝑢2𝛾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (2) 2 Let the vehicle’s configuration be denoted 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 where 𝑄 = R3 × S2 is the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' An example 2D Dubins path (modified with a constant pitch angle to join two altitudes) and a 3D Dubins path that join 𝒒𝑖 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The modified 2D Dubins path uses a constant pitch angle 𝛾𝑐 that is computed from the change in altitude and planar displacement between the start and end configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The modified 2D Dubins path does not satisfy the required pitch angle at the start/end configurations and may violate pitch angle constraints along the path when the change in altitude is large relative to the planar displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Instead, a 3D Dubins path can join two configurations while limiting the pitch angle along the path to within the allowable bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 3D Dubins paths are generated according to [10] by decomposing the 3D path into two decoupled 2D Dubins paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' First, a 2D horizontal Dubins path is constructed in the 𝑥𝑦 plane to join the 2D Dubins configurations (𝑥𝑖, 𝑦𝑖, 𝜓𝑖) and (𝑥 𝑗, 𝑦 𝑗, 𝜓 𝑗) using a horizontal turn radius that is twice the minimum turn radius 𝜌h = 2𝜌min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, a 2D vertical path is constructed, with vertical plane turn radius 𝜌v that is found from [10] 𝜌−2 min = 𝜌−2 h + 𝜌−2 v , (3) to join the 2D Dubins configurations (𝑠𝑖, 𝑧𝑖, 𝛾𝑖) and (𝑠 𝑗, 𝑧 𝑗, 𝛾 𝑗) where 𝑠𝑖 and 𝑠 𝑗 are the initial and final arc-lengths along the Dubins path in the 𝑥𝑦 plane (where 𝑠𝑖 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The turn radii, 𝜌h and 𝜌v, are iteratively varied while satisfying (3) to meet the acceptable pitch angle constraint while minimizing the path length as described in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The length of a 3D Dubins path between two configurations, 𝒒𝑖, 𝒒 𝑗 ∈ 𝑄 is denoted 𝐷(𝒒𝑖, 𝒒 𝑗) : 𝑄2 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Start End Modified 2D Dubins Path 3D Dubins Path Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 2 An example 3D Dubins airplane path (green) [10] joining configurations 𝒒1 = (0, 0, 0, 𝜋 6 , 0) and 𝒒2 = (0, 300 m, 400 m, 0, 0) is compared to a modified 2D Dubins path (red) that join the same pair of locations and heading angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The modified 2D Dubins path is shorter (523 m compared to 1184 m) but violates the pitch angle constraint since a large altitude change is required over a relatively short distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The paths are constructed with the parameters: 𝜌min = 40 m, 𝛾min = −𝜋/12, and 𝛾max = 𝜋/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Environment The airplane operates in an urban environment that consists of a ground plane and a collection of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='5-dimensional objects representing buildings or other structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Let 𝑂 = {𝑂0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑂𝑁𝑂−1} be the set of 𝑁𝑂 objects, where 𝑂𝑖 ⊂ R3 for each 𝑖 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 𝑖th object is an extruded polygon 𝑂𝑖 = {(𝑥, 𝑦, 𝑧) ∈ R3 | (𝑥, 𝑦) ∈ 𝐴𝑖 and 𝑧 ∈ [0, ℎ𝑖]} where 𝐴𝑖 ⊂ 𝑅2 is the object’s footprint and ℎ𝑖 is the height of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The set of points along the boundary of 𝐴𝑖 is a simple two-dimensional polygon denoted 𝜕𝐴𝑖 whose shape is defined by an ordered set of points with a positive signed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Points on the interior of 𝐴𝑖 belong to the set denoted int(𝐴𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The polygonal areas of each object do not intersect int(𝐴𝑖) ∩ int(𝐴 𝑗) = ∅ for all 𝑖 ≠ 𝑗 with 𝑖, 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The height of the tallest object in 𝑂 is denoted ℎmax, and the airplane is constrained to fly in a feasible airspace 𝐹 = 𝐷 × [𝑧min, 𝑧max] − 𝑂 , (4) where 𝐷 ⊂ R2 is the planar region containing the polygonal objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', 𝐴𝑖 ⊂ 𝐷 for all 𝑖 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1}, 𝑧min and 𝑧max > 𝑧min are the minimum and maximum operating altitudes of the airplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The union of all the objects is subtracted from the rectangular volume 𝐷 × [𝑧min, 𝑧max] in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To ensure that 3D Dubins paths joining two configurations does not exceed the feasible airspace or encounter obstacles, the feasible airspace and set of objects can be artificially contracted and inflated, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This work assumes that the minimum altitude 𝑧min is constrained to be above the tallest 3 building, 𝑧min > ℎmax + 2𝜌min, such that the airplane’s feasible airspace is free of objects and there is enough vertical space to maneuver without collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Target Visibility Volumes The airplane is assumed to be equipped with a gimbaled camera and is tasked with inspecting a set of 𝑀 targets located at the points 𝑃 = { 𝒑0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝒑𝑀−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Each target 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) ∈ 𝑃 is located in an unobstructed area of the ground plane or on the exposed surface of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' That is, each target has planar location (𝑝𝑥, 𝑝𝑦) ∈ 𝐷 and altitude 𝑝𝑧 satisfying the following cases: (i) if (𝑝𝑥, 𝑝𝑦) ∩ 𝐴𝑖 = ∅ for all 𝑖 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1} then the target is on the ground plane with 𝑝𝑧 = 0, (ii) if 𝑝𝑦 ∩ 𝜕𝐴𝑖 ≠ ∅ for some 𝑖 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1} then the target is located on the vertical wall of the 𝑖th object and 𝑝𝑧 ∈ [0, ℎ𝑖], or (iii) if 𝑝𝑦 ∩ int(𝐴𝑖) ≠ ∅ then 𝑝𝑧 = ℎ𝑖 such that the target is on top of the 𝑖th object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For each target, a target visibility volume 𝑉𝑖 is defined as the set of points 𝒈 ∈ R3 that have a direct line-of-sight to the target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', not obscured by buildings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Let 𝐿(𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒈, 𝒑) = ( 𝒑 − 𝒈)𝜏 + 𝒈 for 𝜏 ∈ [0, 1] (5) denote a line segment that joints two points 𝒈, 𝒑 ∈ R3 where 𝜏 is a normalized arc-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The visibility volume for a target located at 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) is the subset of the feasible airspace that is within direct line-of-sight to the target, within a maximum range 𝑑max relative to the target, and at least a distance ℎview above the target: 𝑉( 𝒑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝐹, 𝑂, 𝑑max, ℎview) = {𝒈 = (𝑔𝑥, 𝑔𝑦, 𝑔𝑧) ∈ 𝐹 such that || 𝒑 − 𝒈|| ≤ 𝑑max, ℎview + 𝑝𝑧 ≤ 𝑔𝑧 and 𝐿(𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒈, 𝒑) ∩ 𝑂 𝑗 = ∅ for all 𝜏 ∈ [0, 1] and 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑁𝑂 − 1}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (6) For brevity, visibility volumes (6) are henceforth denoted 𝑉( 𝒑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The maximum range 𝑑max constraint models minimum image resolution requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The minimum height-above-target ℎview < 𝑑max constraint ensures images are captured with sufficient surrounding context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', the point target may actually represent an extended body that should be contained in the image) or to reduce gimbal pointing speed and precision requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For the problem to be well posed, there should always exist at least one valid viewing point above each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This condition may be satisfied by the following parameter constraints: 𝑧min ≤ ℎview + ℎmax ≤ 𝑧max , (7) 𝑧min ≤ 𝑑max , (8) 2𝑑max < || 𝒑𝑖 − 𝒑 𝑗|| for all 𝒑𝑖, 𝒑𝑖 ∈ 𝑃 with 𝒑𝑖 ≠ 𝒑 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (9) If a target is located on top of the highest object, then constraint (7) ensures that a viewing point exists that is below the maximum feasible altitude and above the minimum feasible altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For targets that are located on the ground plane, constraint (8) ensures that the sensor range is sufficiently large to view the target from the minimum feasible altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Lastly, constraint (9) is a simplifying assumption that guarantees targets are spaced sufficiently far apart such that their visibility volumes do not intersect 𝑉( 𝒑𝑖) ∩ 𝑉( 𝒑 𝑗) = ∅ for all 𝑖, 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑀 − 1} with 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' View-planning Problem Statement Let 𝐵(𝒒) be a mapping from a configuration 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 to an integer in the set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑀 − 1} that identifies the visibility volume corresponding to 𝒒, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', the integer 𝐵(𝒒) corresponds to the target 𝒑𝐵(𝒒) ∈ 𝑃 for which (𝑥, 𝑦, 𝑧) ∈ 𝑉( 𝒑𝐵(𝒒)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' If 𝒒 is not contained in any visibility volume then 𝐵(𝒒) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The optimization problem is to find the sequence of vehicle configurations 𝒒0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝒒𝑀−1 that minimize 𝑀−1 ∑︁ 𝑖=0 𝐷(𝒒𝑖, 𝒒𝑖+1) + 𝐷(𝒒𝑀−1, 𝒒0) , (10) subject to 𝐵(𝒒𝑖) ≠ 𝐵(𝒒 𝑗), for all 𝑖, 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑀 − 1} with 𝑖 ≠ 𝑗 , (11) 𝐵(𝒒0) ∪ · · · ∪ 𝐵(𝒒𝑀−1) = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑀 − 1} , (12) where the cost function (10) is the total length of the 3D Dubins paths in the tour, the constraint (11 ensures that each vehicle configuration lies within a unique visibility volume and the constraint 12) ensures that all visibility volumes are visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The view planning problem (10)–(12) is a mixed continuous/combinatorial optimization problem with a nonlinear cost function and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Since the vehicle travels at a constant speed the minimum-length tour is also the minimum-time tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 2D Algorithms In this section, a target visibility volume mesh approximation is described (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A) followed by a description of two-dimensional algorithms (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='B and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='C) that solve the view planning problem (10)–(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithms discussed here include (i) traveling directly over each target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', formulating a Dubins traveling salesperson problem (DTSP) [12]), and (ii) the DTSP with neighborhoods (DTSPN) to visit one point in a set of visibility polygons corresponding to the targets [6] that is modified to use an optimized altitude for defining the visibility polygons A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Target Visibility Volume Approximation Volumes in 3D are commonly approximated by a triangular mesh [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' While many prior works on the DTSP have assumed simplified 3D geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', spheres, cylinders), we propose to use triangular meshes since they can represent arbitrary geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 𝑖th target visibility volume 𝑉𝑖 is approximated with 𝑁𝐹 triangular mesh elements resulting in the mesh ˆ𝑉𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Let | ˆ𝑉𝑖| denote the total number of mesh elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 𝑗th mesh element in ˆ𝑉𝑖 is defined as a set of vectors ˆ𝑉𝑖 𝑗 = {𝒄0 𝑖 𝑗, 𝒄1 𝑖 𝑗, 𝒄2 𝑖 𝑗, 𝒏𝑖 𝑗} where the vectors 𝒄0 𝑖 𝑗, 𝒄1 𝑖 𝑗, 𝒄2 𝑖 𝑗 ∈ R3 are the positions of the vertices of a triangular mesh element, and 𝒏𝑖 𝑗 ∈ R3 is an outward pointing normal vector, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The mesh-based target visibility volumes ˆ𝑉𝑖 are computed using the painter’s algorithm [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A sphere centered on each target is decomposed into six mutually perpendicular views, and each view looks out from the target point location with a 90-degree field-of-view thereby covering one of the six sides of a cube enclosing the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' OpenGL [15] and a special version of the geometric depth map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', inverse depth, is used to capture the depth of scene objects in the direction of each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' After calculating the depth values, those that are less than or equal to 𝑑max are tessellated into a preliminary 3D visibility volume mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This mesh is genus-0 [13], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', a deformation of the sphere, and is also a manifold surface amenable to constructive solid geometry (CSG) Boolean operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, the mesh is intersected with the feasible airspace 𝐹 and the minimum viewing distance constraint ℎmin is imposed using CSG Boolean intersection operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To reduce the number of vertices in the resulting mesh a decimation procedure is applied [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 3 Example target visibility region with a mesh element defined by three vertices 𝒄0, 𝒄1, 𝒄2 and outward pointing normal vector 𝒏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Baseline Algorithm: Dubins Traveling Salesperson Problem (DTSP) The DTSP is the problem of finding the shortest planar tour that visits all points in a graph once using points that are connected with 2D Dubins paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Since the objects considered here are extruded polygons, there are no features that can block viewing targets from above (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', bridges or tunnels are not admissible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Consequently, the view planning problem (10)–(12) can be solved with the DTSP by flying at a fixed altitude directly over each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' All feasible altitudes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', that are common to all visibility volumes) lead to identical cost tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To account for the different possible heading angles at each overhead location the heading-angle-discretized DTSP is adopted [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' An example solution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Optimized Altitude DTSP with Neighborhoods (DTSPN) A more sophisticated approach developed by Obermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' [6] considers the fixed altitude slices of the target visibility volumes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', planar visibility polygons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Vehicle configurations in each visibility polygon are sampled and a DTSPN [5, 6] is formulated to visit one configuration in each visibility polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In [6], two sampling algorithms were proposed: entry pose sampling—wherein samples are made along the edge of the polygon with heading angles that are tangent or inward pointing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 4b)—and interior pose sampling—wherein samples are placed uniformly in a grid on the interior of the visibility polygons with uniformly sampled heading angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In [6], entry pose sampling gave lower cost solutions than interior pose sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Thus, the entry pose sampling method is adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The constraints of the view planning problem (7)–(9) allow for visibility volumes to occupy disjoint segments of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' That is, there may not exist an altitude 𝑧∗ ∈ [𝑧min, 𝑧max] that is common to all visibility volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' While this does not pose an issue for some of the 3D algorithms proposed later, these cases cannot be solved by the 2D (constant-altitude) algorithms described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' However, introducing the additional constraint ℎmax + ℎview ≤ 𝑑max (13) ensures that the visibility volumes for a target located on the ground plane and for a target located atop the highest object have at least one common altitude at 𝑧∗ = 𝑑max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In general, there is a range of admissible altitudes 𝑧∗ that may be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The choice of altitude impacts the 2D DTSPN algorithm since visibility polygons change in shape and size as the altitude varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Intuitively, larger polygons are preferred over smaller ones since this increases the set of candidate configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This work proposes to identify an optimal working altitude for the 2D algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' First, 𝑛slice polygons are generated from each visibility volume mesh (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', for all targets) using the method described in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Let P = polygonFromMesh( ˆ𝑉, 𝑧) denote the polygon that results from slicing mesh ˆ𝑉 at altitude 𝑧 and let polygonArea(P) denote the corresponding area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The optimal altitude 𝑧∗ is chosen as the one that maximizes the sum of polygonal areas across all visibility polygons: 𝑧∗ = argmax 𝑧∈𝑍 𝑀−1 ∑︁ 𝑖=0 polygonArea(polygonFromMesh( ˆ𝑉𝑖, 𝑧)) (14) where 𝑍 = {𝑧0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑧𝑛slice−1} is a set of altitudes at which the polygons are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Note that all altitudes 𝑧 ∈ 𝑍 are constrained such that 𝑧min ≤ 𝑧 ≤ 𝑧max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Also, note that 𝑧0 and 𝑧slice−1 are not 𝑧min and 𝑧max respectively, instead 𝑧0 and 𝑧slice−1 are slightly offset into the body to avoid floating point error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The selection of 𝑧∗ is visualized in Fig 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (a) 2D Dubins traveling salesperson problem (DTSP) (b) 2D DTSP with neighborhoods (DTSPN) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 4 Example solutions to the view-planning problem using 2D constant-altitude algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The green regions are visibility polygons for a chosen altitude 𝑧∗ while the black arrows represent the heading angle at sampling points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The multicolored line is the solution path of the TSPs with increasing path length represented by color changes from red to purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The DTSP (a) is solved with entry pose sampling using eight headings samples directly over the targets, while the 2D DTSPN (b) uses eight sample locations around the perimeter of the polygon with four heading angles that are tangent or point into the corresponding visibility polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 3D Algorithms Inspection tours that admit three-dimensional maneuvering can potentially lead to path length reductions when compared to two-dimensional (constant-altitude) tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The solution techniques in this work all use a transformation 6 approach to solve the (2D or 3D) DTSPN according to the following steps: compute the approximation of the target visibility volume (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A), sample the visibility volumes to create graph vertices corresponding to vehicle configurations, calculate edge costs between vertices using the 3D Dubins path planning algorithm, and solve for a DTSPN tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A details three algorithms to sample the target visibility volumes: random face sampling, 3D edge sampling, and global weighted face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='2 then introduces a heuristic approach that improves the edge cost computation time using a modified Euclidean distance edge cost and a geometric approach to assign heading and pitch angles at each configuration in the tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' (a) Optimized altitude with entry pose sampling (b) Random face sampling (c) 3D edge sampling (d) Global weighted face sampling Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 5 Visualizations of the four different sampling algorithms: optimized altitude with entry pose sampling, random face sampling, 3D edge sampling, and global weighted face sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Two-dimensional representations of the visibility volumes are in gray, altitude slices are orange lines, and sampled configurations are blue circular markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Sampling Strategies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Random Face Sampling The random face sampling algorithm extends the 2D entry pose strategy from [6] to sample 3D vehicle configurations across the surface of the target visibility volume with a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The approach is detailed in Algorithm 1 and visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithm randomly finds 𝑛pts three-dimensional points on the faces of each triangular mesh in the set of triangular meshes ˆ𝑉0:𝑀−1 = { ˆ𝑉1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , ˆ𝑉𝑀−1} and assigns to each point a set of configurations with 𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle configurations per visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' First, the set of configurations Q is initialized as an empty set, and the area of each face in the mesh is calculated (lines 3-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The area of each triangular face element, 𝑎𝑖 𝑗, is calculated by the elementArea function using 𝑎𝑖 𝑗 = elementArea( ˆ𝑉𝑖 𝑗) = 1 2 ||(𝒄0 𝑖 𝑗 − 𝒄1 𝑖 𝑗) × (𝒄0 𝑖 𝑗 − 𝒄2 𝑖 𝑗)|| , (15) where × is the vector cross product and 𝒄0 𝑖 𝑗, 𝒄1 𝑖 𝑗, and 𝒄2 𝑖 𝑗 are the three vertices contained in the triangular face element ˆ𝑉𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, the proportion of each face area to the total surface area of the mesh ˆ𝑉𝑖 is calculated using element-wise division (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The randomSetOfIndices function identifies 𝑛pts random faces by sampling faces with probability in proportion to the weights 𝒘𝐹 (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The use of the proportional surface area during the random selection process gives every point on the surface of the target visibility region an equal chance of being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For each selected triangular face, a point on the face is randomly selected using a Barycentric coordinate system [18] (lines 10-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The Barycentric coordinate system allows for the mapping of two random numbers 𝑟0 and 𝑟1 sampled uniformly from the interval [0, 1] onto a triangle, embedded in R3, with the weighted sum of its vertices [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The random numbers 𝑟0 and 𝑟1 are first sampled (line 11) and then a position on the chosen triangular face is determined (line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For each position, 𝑛𝜓 heading angles sampled uniformly between 0 and 2𝜋 as well as 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled uniformly then added to the set of vehicle configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The runtime of the algorithm is dominated by the nested for loops on lines 2 and 4 running 𝑀| ˆ𝑉|max times—where | ˆ𝑉|max = max𝑖∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=',𝑀−1}(| ˆ𝑉𝑖|) is the maximum number of 7 faces in a single mesh—and the collections of nested loops on lines 13-14 which run 𝑀𝑛pts𝑛𝜓𝑛𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' When the number of mesh faces in a visibility volume is greater than the number of samples collected | ˆ𝑉|max > 𝑛pts𝑛𝜓𝑛𝛾 the runtime is 𝑂(𝑀| ˆ𝑉|max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Algorithm 1 Random Face Sampling function: RandomFaceSampling( ˆ𝑉0:𝑀−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛pts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛𝜓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛𝛾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝛾max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝛾min) input: target visibility volume mesh ˆ𝑉0:𝑀−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of points to sample 𝑛pts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of heading angles 𝑛𝜓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of pitch angles 𝑛𝛾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' max pitch angle 𝛾max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' min pitch angle 𝛾min output: a set of vehicle configurations for each target Q 1: Q ← ∅ 2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do 3: Q𝑖 ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒂𝑖 ← ∅ 4: for ˆ𝑉𝑖 𝑗 ∈ ˆ𝑉𝑖 do 5: 𝒂𝑖 ← 𝒂𝑖 ∪ elementArea( ˆ𝑉𝑖 𝑗) 6: end for 7: 𝒘𝐹 = 𝒂𝑖/�|𝒂𝑖 |−1 𝑗=0 𝑎𝑖 𝑗 8: 𝐼 ← randomSetOfIndices(𝑛pts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒘𝐹) 9: for 𝑖 ∈ 𝐼 do 10: 𝒄0 𝑖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒄1 𝑖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝒄2 𝑖 𝑗 ← getVertices( ˆ𝑉𝑖 𝑗) 11: 𝑟0 ∼ U[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑟1 ∼ U[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='1] 12: 𝒔 ← 𝒄0 𝑖 𝑗 (1 − √𝑟0) + 𝒄1 𝑖 𝑗 √𝑟0(1 − 𝑟1) + 𝒄2 𝑖 𝑗 √𝑟0𝑟1 13: for 𝑗 ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝜓 − 1} do 14: for 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝛾 − 1} do 15: Q𝑖 ← Q𝑖 ∪ �𝒔, 2𝑗𝜋/𝑛𝜓, 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1)� 16: end for 17: end for 18: end for 19: Q ← Q ∪ Q𝑖 20: end for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 3D Edge Sampling The second sampling strategy proposed is 3D edge sampling wherein the 2D entry pose strategy from [6] is extended to sample 3D vehicle configurations across the lowest feasible altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For the visibility volume shapes studied here this is also the altitude where the cross-sectional area is largest for each shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 3D edge sampling algorithm, detailed in Algorithm 2 and visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 5c, finds 𝑛pts three-dimensional points on the polygon created by slicing the triangular mesh along the lowest feasible altitude and distributing points uniformly along the perimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithm then assigns a set of configurations to each point with 𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle configurations per visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' First, the set of configurations Q is initialized as an empty set (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Then, for each visibility volume a subset of points contained in that volume is initialized (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, the 𝑧 minimum altitude for the triangular mesh is found by finding the minimum height coordinate in the set of vertices in the mesh ˆ𝑉 𝑧 𝑖 (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' After, the polygonFromMesh algorithm takes the triangular mesh and the 𝑧min altitude and returns a polygonal slice of the mesh (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A set of points, 𝝀 ∈ R2, placed uniformly along the edge of the polygon is found using the uniformPerimeterPoints which takes a polygon and the number of points desired as arguments (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Note that lines 4–6 can be modified to produce samples at multiple altitude slices if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, the algorithm iterates through each sampled point and assigns heading and pitch angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To ensure inward-pointing heading angles, the direction of the line segment containing the sample point is found using the tangentAngle function (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The points in the polygon defined by polygonFromMesh have a positive signed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Thus, the inward-pointing heading angles are the angles from [0, 𝜋] measured counter-clockwise from the tangent angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For each position, 𝑛𝜓 heading angles between 𝜓𝑞 and 𝜓𝑞 + 𝜋 and 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled uniformly and returned as part of the vehicle configurations (lines 9-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To achieve 𝑛 equally spaced angle samples, including the minimum and maximum angle, the range is divided into 𝑛 − 1 sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The max function, on lines 10 and 12, ensures the range is never divided by zero (the case where 𝑛𝛾 or 𝑛𝜓 is one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The runtime complexity is dominated 8 by the for loops on lines 2-18 which have a worst-case running time complexity of 𝑂(𝑀𝑘), where 𝑘 = | ˆ𝑉|2 max + 𝑛pts𝑛𝜓𝑛𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' | ˆ𝑉|2 max is the runtime of the polygonFromMesh algorithm while 𝑛pts𝑛𝜓𝑛𝛾 is the runtimes for the nested for loops (lines 9-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater than the total number of configurations returned, | ˆ𝑉|2 max > 𝑛pts𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀| ˆ𝑉|2 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Algorithm 2 3D Edge Sampling function: 3DEdgeSampling( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝛾max, 𝛾min) input: target visibility volume mesh ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of pitch angles 𝑛𝛾, max pitch angle 𝛾max, min pitch angle 𝛾min output: a set of vehicle configurations for each target Q 1: Q ← ∅ 2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do 3: Q𝑖 ← ∅ 4: 𝑧min ← min( ˆ𝑉 𝑧 𝑖 ) 5: P ← polygonFromMesh(𝑧min, ˆ𝑉𝑖) 6: {𝝀0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝝀𝑛pts−1} ← uniformPerimeterPoints(P, 𝑛pts) 7: for 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛pts − 1} do 8: 𝜓𝑞 ← tangentAngle(𝝀𝑚, P) 9: for 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝜓 − 1} do 10: 𝜓 ← 𝜓𝑞 + 𝑗𝜋/max(𝑛𝜓 − 1, 1) 11: for 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝛾 − 1} do 12: 𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1) 13: Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧min, 𝜓, 𝛾) 14: end for 15: end for 16: end for 17: Q ← Q𝑖 ∪ Q 18: end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Global Weighted Face Sampling The third proposed sampling strategy is global weighted face sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Rather than sampling the visibility volumes at the lowest altitude, all target visibility volumes are sampled along a common set of altitude planes and the number of samples allocated to each plane is determined by the cross-sectional perimeter distribution of each altitude summed across all target visibility volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This approach places more samples at altitudes common to all targets that, on average, also have large cross-sectional areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This sampling method is detailed in Algorithm 3 and visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithm takes a set of target visibility meshes ˆ𝑉0:𝑀−1 and returns a set of vehicle configurations Q for each mesh given the parameters 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝑛slice, 𝛾max, and 𝛾min where 𝑛slice ≥ 2 is the number of altitude slices to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Let ˆ𝑉 𝑧 0:𝑀−1 denote the set of all 𝑧 heights for every vertex contained across the 𝑀 meshes ˆ𝑉0:𝑀−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' First, the global minimum, the global maximum altitude, and the slicing altitude step size are found (lines 1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Then a vector 𝝁 is initialized with zeros, denoted as 0𝑛slice×1 (line 3), and later stores the total perimeter summed across all visibility polygons at the corresponding altitude slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The target visibility volumes are sliced into polygons with fixed altitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' parallel to the 𝑥𝑦 plane) using the polygonFromMesh function, lines 4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The lowest 𝑧 plane is the visibility volumes’ global minimum 𝑧 height (𝜁min) and the highest 𝑧 plane is the visibility volumes’ global maximum 𝑧 height (𝜁max), line 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The nominal set of altitude planes is then 𝑍 = {𝑧0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑧𝑛slice−1} where 𝑧0 = 𝜁min, 𝑧𝑛slice−1 = 𝜁max and 𝑧𝑖+1 − 𝑧𝑖 = 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' At each plane 𝑧 ∈ 𝑍, polygons are created from the target visibility volume and the polygons’ perimeters are accumulated, line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The sample points in each 𝑧 plane are then distributed in proportion to the accumulated perimeters, lines 11-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The function iteratePerimeters takes six arguments: the mesh to iterate across, a perimeter distribution, the minimum altitude, the maximum altitude, the step size, and the total number of sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' It returns a variable number of 𝑛𝑧 ≤ 𝑛slice elements where each element is a pair consisting of a 𝑧𝑟 altitude and the number of points to sample at that altitude, 𝑛𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' An altitude slice 𝑧𝑟 is either an element of 𝑍 and/or an altitude located at the top or bottom of each visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' At each altitude 𝑧𝑟 the corresponding value of 𝝁 is determined (or interpolated, in the special case that 𝑧𝑟 ∉ 𝑍) and the 𝑛pts are distributed to each 𝑧𝑟 in proportion to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In the event that no slices intersect the visibility mesh 9 then 𝑛𝑧 = 2 and the heights 𝑧𝑟 are returned corresponding to the top and bottom of the target visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Next, samples 𝝀 ∈ {𝝀0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', 𝝀𝑛𝑟−1} are placed uniformly around the perimeter of each polygon created by the intersection of the 𝑧𝑟 planes and the target visibility volume with the function uniformPerimeterPoints, line 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The heading and pitch angles are sampled in the same way as entry pose sampling [6], pointing tangent or inward with respect to the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The angle tangent to each point 𝝀 on the perimeter of the polygon is found with the tangentAngle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The pitch angles are uniformly sampled within the pitch angle constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The runtime complexity is dominated by the for loops on lines 11-29 which have a worst-case running time complexity of 𝑂(𝑀𝑛slice𝑘), where 𝑘 = | ˆ𝑉|2 max + 𝑛𝑟𝑛𝛾𝑛𝜓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater than the total number of configurations returned, | ˆ𝑉|2 max > 𝑛𝑟𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀𝑛slice| ˆ𝑉|2 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Algorithm 3 Global Weighted Face function: GlobalWeightedFace( ˆ𝑉0:𝑀−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛pts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛𝜓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛𝛾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑛slice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝛾max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝛾min) input: set of triangular meshes ˆ𝑉0:𝑀−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of points to sample 𝑛pts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of heading angles 𝑛𝜓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of pitch angles 𝑛𝛾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' number of altitude slices 𝑛slice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' max pitch angle 𝛾max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' min pitch angle 𝛾min output: a set of vehicle configurations for each target Q 1: 𝜁max ← max( ˆ𝑉 𝑧 0:𝑀−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝜁min ← min( ˆ𝑉 𝑧 0:𝑀−1) 2: 𝐿 ← (𝜁max − 𝜁min)/(𝑛slice − 1) 3: 𝝁 ← 0𝑛slice×1 4: for 𝑖 ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛slice − 1} do 5: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do 6: P ← polygonFromMesh(𝜁min + 𝐿𝑖, ˆ𝑉𝑖) 7: 𝜇𝑖 ← 𝜇𝑖 + perimeter(P) 8: end for 9: end for 10: Q ← ∅ 11: for ˆ𝑉𝑖 ∈ ˆ𝑉1:𝑀 do 12: (𝑧𝑟, 𝑛𝑟)𝑛𝑧−1 𝑟=0 ← iteratePerimeters( ˆ𝑉𝑖, 𝝁, 𝜁min, 𝜁max, 𝐿, 𝑛pts) 13: Q𝑖 ← ∅ 14: for 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝑧 − 1} do 15: P ← polygonFromMesh(𝑧𝑟, ˆ𝑉𝑖) 16: {𝝀0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝝀𝑛𝑟−1} ← uniformPerimeterPoints(P, 𝑛𝑟) 17: for 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝑟 − 1} do 18: 𝜓𝑞 ← tangentAngle(𝝀𝑚, ˆ𝑉𝑖) 19: for 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝜓 − 1} do 20: for 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' , 𝑛𝛾 − 1} do 21: 𝜓 ← 𝜓𝑞 + 𝑘𝜋/max(𝑛𝜓 − 1, 1) 22: 𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1) 23: Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧𝑟, 𝜓, 𝛾) 24: end for 25: end for 26: end for 27: end for 28: Q ← Q𝑖 ∪ Q 29: end for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Proposed Heuristics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Modified Euclidean Distance Traveling Salesperson Problem with Neighborhoods (METSPN) A bottleneck in the 3D DTSPN algorithms is the computation of the edge costs that require solving for a 3D Dubins path between two configurations 𝒒𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖, 𝜓𝑖, 𝛾𝑖) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Since the Dubins path is asymmetric the corresponding edge cost must be computed for each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Here, we propose an approximation to 10 this edge cost ˆ𝐷(𝒒𝑖, 𝒒 𝑗) = max � |𝛿𝑧| sin 𝛾limit , ∥𝒔𝑖 − 𝒔 𝑗 ∥2 � , (16) where 𝛿𝑧 = 𝑧 𝑗 − 𝑧𝑖, 𝛾limit = 𝛾max if 𝛿𝑧 > 0 and 𝛾limit = 𝛾min otherwise, 𝒔𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and 𝒔 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The calculation is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The distance (16) is a lower bound on the actual 3D Dubins path length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 6 Visualization of the modified Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The Euclidean distance shown in blue with a pitch angle 𝛾 > 𝛾limit is modified by extending the distance traveled in the 𝑥𝑦 plane resulting in the red line with pitch angle 𝛾limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' ˆ𝐷𝑥𝑦 refers to the length of the Dubins path projected onto the 𝑥𝑦 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' ˆ𝐷(𝒒0, 𝒒1) ≤ 𝐷(𝒒0, 𝒒1), and is significantly faster to compute than solving for the Dubins path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Using this edge cost leads to a variant of the DTSPN we refer to as the modified Euclidean distance traveling salesperson problem (METSPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Solving the METSPN gives a tour of 3D locations to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Once a tour is found for the METSPN it is converted into a feasible sequence of Dubins paths by assigning heading and pitch angles as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Bisecting Angle Approximation To assign heading and pitch angles a heuristic is adopted that extends the mean angle algorithm developed in [19] to three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The approach is summarized in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The proposed bisecting angle approximation takes as parameters: 𝑽 a 𝑀 × 3 matrix corresponding to the sequence of vertices in the METSPN tour and the problem parameters: 𝜌min, 𝛾min, and 𝛾max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithm returns a set of vehicle configurations Q at each point in 𝑽 with heading and pitch angles defined as the angle bisector of each consecutive triplet of vertices (for points spaced far apart) or as a straight segment (for points spaced close together).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To obtain the angle bisector at each vertex, calculate vectors from the preceding vertex 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 = (𝑢𝑥, 𝑢𝑦, 𝑢𝑧) and to the following vertex 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 = (𝑤𝑥, 𝑤𝑦, 𝑤𝑧) (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The vector 𝒃 = 𝒘 + 𝒖 = (𝑏𝑥, 𝑏𝑦, 𝑏𝑧) determines the heading angle 𝜓 in the 𝑥𝑦 plane computed with the four-quadrant arctangent function (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A visualization of the calculation can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The circular indexing of 𝑽, a 𝑀 by 3 matrix, allows for the index −1 to refer to the last Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 7 The notation used to determine the bisector vector for a triplet of three points: 𝑽𝑖−1, 𝑽𝑖, 𝑽𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The orientation of the vectors 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 and 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 are summed and normalized resulting in the vector 𝒗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The heading angle 𝜓 is the component of 𝒃 in the 𝑥𝑦 plane while the pitch angle 𝛾 is measured from the 𝑥𝑦 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' column of 𝑽 and the index 𝑛 to refer to the first element of 𝑽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The pitch angle bisector is the angle between the vector 𝒃 11 and the 𝑥𝑦 plane (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The resulting angle is saturated to be within the pitch angle bounds on line 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' If vertices are close together then curve-curve-curve (CCC) Dubins paths may be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This should be avoided because the cost of (CCC) Dubins paths is much greater than the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The likelihood of CCC paths occurring is reduced by setting the heading and pitch in the direction of the line between two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' If the distance between two vertices is small (less than the long path case in [20]), then heading and pitch angles are aligned with the while loop on lines 10-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To align the headings of two configurations, the vector between the internal coordinates is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The angle of this vector, 𝒘, about the 𝑧 axis is used as the heading angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Then, the angle between the 𝑥𝑦 plane and the vector 𝒘 is found and saturated between 𝛾min and 𝛾max to set the pitch angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Inside the loop, the index is advanced once but it is also advanced a second time if the current vertex and the next vertex are within 4𝜌min units of each other (worst case for the long path case [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The second index advance is required to pass over the next configuration because it was just modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Algorithm 4 Bisect Angle Approximation function: BisectAngleApprox(𝑽, 𝜌min, 𝛾min, 𝛾max) input: 𝑽 is a 𝑛 by 3 matrix of vertices that solve the METSPN, minimum turn radius 𝜌min, minimum pitch angle 𝛾min, maximum pitch angle 𝛾max output: set of configurations solving a DTSP Q 1: Q ← ∅ 2: for 𝑖 ∈ {0, 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 𝑀 − 1} do 3: 𝒃 ← 𝑽𝑖+1 + 𝑽𝑖−1// indexing into 𝑽 is circular 4: 𝜓 ← atan2(𝑏𝑥, 𝑏𝑦) 5: 𝛾 ← atan2(𝑏𝑧, √︃ 𝑏2𝑥 + 𝑏2𝑦) 6: 𝛾 ← max(min(𝛾, 𝛾max), 𝛾min) 7: Q ← Q ∪ (𝑽𝑖, 𝜓, 𝛾) 8: end for 9: 𝑖 ← 0 10: while 𝑖 < |𝑽| do 11: if ||𝑽𝑖 − 𝑽𝑖+1|| < 4𝜌min then 12: 𝒘 ← 𝑽𝑖+1 − 𝑽𝑖 13: 𝜓 ← atan2(𝑢𝑥, 𝑢𝑦,) 14: 𝛾 ← atan2(𝑢𝑧, √︃ 𝑢2𝑥 + 𝑢2𝑦) 15: 𝛾 ← max(min(𝛾, 𝛾max), 𝛾min) 16: Q𝑖𝜓 ← 𝜓, Q(𝑖+1) 𝜓 ← 𝜓 17: Q𝑖𝛾 ← 𝛾, Q(𝑖+1)𝛾 ← 𝛾 18: 𝑖 ← 𝑖 + 1 19: end if 20: 𝑖 ← 𝑖 + 1 21: end while C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Illustrative Examples An example of a view planning solution for five targets scattered around a city model of Charlotte, North Carolina is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The example was constructed assuming a Dubins airplane model having a curvature radius of 𝜌min = 40 m and pitch angle constraints 𝛾 ∈ [−𝜋/12, 𝜋/9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The random-face algorithm was used with 𝑛pts = 8 samples per visibility volume, 𝑛𝜓 = 4 heading angles per sample, and 𝑛𝛾 = 1 pitch angle per sample-heading angle pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The visibility volumes for targets that had no occlusions had a common dome shape, whereas targets located closer to objects had more arbitrary shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Another example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 8b illustrates the solution for five targets in a model of New York City, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This example compares the three-dimensional random-face algorithm with 𝑛pts = 32, 𝑛𝜓 = 8, and 𝑛𝛾 = 3 pitch angles, to the two-dimensional optimized altitude entry pose sampling algorithm with 𝑛pts = 32, 𝑛𝜓 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 3D path can change altitude which allowed the algorithm to find a lower cost path of 3920m while the 2D algorithm maintained constant altitude and found a path of cost 4285m, a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='9% reduction in path cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 12 (a) 3D DTSP with neighborhoods (DTSPN) in Charlotte, North Carolina (b) 3D DTSP with neighborhoods (DTSPN) in New York City, New York Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 8 Solutions to the 3D Dubins traveling salesperson problem with neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Panel (a) was computed using the random face algorithm in light blue with 8 samples per target visibility volume, four heading angles per sample, and one pitch angle per sample-heading angle pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Panel (b) was computed using the random face sampling algorithm in dark blue with 𝑛pts = 32 samples per target visibility volume, 𝑛𝜓 = 8 heading angles per sample, and 𝑛𝛾 = 3 pitch angles per sample-heading pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' the two-dimensional entry pose sampling from [6] in magenta with 𝑛pts = 32 samples per target visibility volume and 𝑛𝜓 = 4 heading angles per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The target visibility volume is translucent white with black edges and the targets are red spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The green spheres are the vehicle configurations for the solution to the DTSPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The environment shown is a section of New York City, New York obtained from the OpenStreetMap database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Building heights are indicated by the varying color scale from yellow to purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Numerical Performance Study The 2D algorithms from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III were compared to the 3D algorithms from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV through a Monte-Carlo experiment that randomized target locations and a number of targets located in an urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This section describes the implementation of the algorithms, the design of the Monte-Carlo study, and discusses the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Implementation The algorithms in this work were written in python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='9 ∗[21] using a number of packages, including Shapely [22] for polygonal operations and NumPy [23] for working with matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The GLKH traveling salesperson solver [24] was used to solve the generalized traveling salesperson problems that arise from DTSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The target visibility volumes were created with data from OpenStreetMap [25], inverse depth calculations from the target location using OpenGL [15], and Blender [26] was used for intersecting the triangular meshes within the feasible airspace 𝐹 as well as decimating the meshes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', reducing the number of triangular faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This work uses [27] to compute 2D Dubins paths for the 2D algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithm simulations were performed on an AMD Threadripper 3990X running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='04 with one thread allocated to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Monte-Carlo Experiment A Monte-Carlo experiment was designed using the environments described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The environments were created by capturing all of the buildings in a rectangular area in New York City with the OpenStreetMap database and limiting the building heights to 300 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Target locations were randomized for each trial and determined by sampling the environment and placing targets on the ground, the wall of buildings, or the roofs of buildings according to a user-defined distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The radius of the target visibility volumes was 300 m with each target being at least 600 m apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The proposed sampling methods and heuristics are independent and studied here in different combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithms parameters were varied as follows: the number of samples per visibility volume was varied between 𝑛pts = {2, 4, 8, 16, 32}, the number of heading angles per sample was 𝑛𝜓 = {2, 4, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To reduce the number of trials, only one pitch angle (𝑛𝛾 = 1) of 0◦ was used by passing 0◦ for 𝛾min and 𝛾max to the random face sampling (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='1), 3D edge sampling (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='2), and global weighted face sampling (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='3) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The Dubins airplane had ∗The implementation of this study can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='com/robotics-uncc/VisualTour3DDubins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 13 Table 1 Description of environments obtained from an OpenStreetMap database for New York City, USA, and used for the Monte-Carlo experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Number of Targets Number of objects Width Depth 5 5624 1986 m 2090 m 10 9202 2809 m 2857 m 15 11584 3440 m 3621 m 20 12119 3972 m 4181 m a minimum curvature radius of 𝜌min = 40 m and a pitch angle constrained between -𝜋/12 and 𝜋/9, similar to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A total of 80 configurations of targets were generated, divided evenly among groups of 5, 10, 15, and 20 targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Every combination of algorithm parameters was evaluated with the 80 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The normalized tour cost (total length of the tour divided by the turn radius) and the computation time were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The algorithms are denoted by acronyms wherein the prefix is either 2D-DTSP, 2D-DTSPN, 3D-DTSPN, or 3D-METSPN corresponding to the algorithms of Sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='B, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='C, IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 2D-DTSP is followed by a dash and an integer representing the number of heading angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The remaining two algorithms are described by a sampling method acronym: entry pose sampling (ETRY) from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='C, random face sampling (RFAC) from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='1, 3D edge sampling (E3D) from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='2, or global weighted face sampling (GWF) from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='3 followed by a dash and an integer representing the number of heading angles and another dash and an integer representing the number of samples per target visibility volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', 2D-DTSPN-ETRY-4-16 corresponds to a 2D DSTPN using entry pose sampling with 4 heading angles and 16 sample points per target visibility region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Analysis of Monte-Carlo Study In general, two-dimensional methods at a fixed altitude performed better if the targets are all located at similar heights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' whereas, 3D methods trended towards better tour cost when targets occupy a wide range of altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The median path length, normalized by dividing the cost by the minimum curvature radius, of each view-planning tour (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', cost) of the Monte-Carlo runs for an increasing number of targets, the number of heading angles is held at 𝑛𝜓 = 8, and the number of pitch angles is held at 𝑛𝛾 = 1 is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The DTSP algorithms that only visit a single point (gray) have one location sample per visibility volume but the lines were extended along the abscissa for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The DTSP is inefficient in our problem because shorter paths can be obtained between targets by flying through the boundary of their corresponding visibility volumes rather than requiring the paths to pass through the visibility volume centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' As the number of heading angles increases the mean cost of the solution decreases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The METSPN algorithms have a similar cost to the eight sampled heading angle solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Most of the medians for different algorithms approach an asymptote, suggesting that they are converging towards a fixed median tour cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', further increasing the number of samples has diminishing returns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' For a large number of samples, the proposed random face sampling algorithm yields a lower tour cost than the optimized altitude 2D algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' However, the median of the 3D edge sampling algorithm is less than the optimal altitude 2D algorithm for all numbers of samples greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This may be due to the 3D algorithms spreading their samples across another dimension (altitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 3D algorithms that spread the samples along the vertical dimension of each visibility volume perform worse than the algorithm that only samples one altitude slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This suggests that distributing the points in the horizontal plane is more important than distributing them in the vertical direction for this particular environment and visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The sensor model creates visibility volumes with the most horizontal variation at the bottom of the shape as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' therefore, sampling the visibility volumes at the bottom is the best way to produce samples with the greatest horizontal variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' To isolate the effects of the different sampling methods, the results are examined for the case where the number of samples is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8, and the number of pitch angles is 𝑛𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Box plots of those trials can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The medians of the 3D methods (black bar in the middle of the colored box) are lower than the medians of the 2D methods suggesting that the 3D methods are able to more consistently find lower-cost solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The difference between medians of 2D and 3D methods grows as the number of target visibility volumes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The range of solutions for the different methods, denoted by the vertical black bars, is large and suggests that the difference between the solutions produced by the 2D and 3D cases is variable and sensitive to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The time for each algorithm to execute on a single thread is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' It can be seen that 14 2 4 8 16 32 Samples per Target 170 180 190 200 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 5 Targets 2 4 8 16 32 Samples per Target 225 250 275 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 10 Targets 2 4 8 16 32 Samples per Target 270 300 330 360 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 15 Targets 2 4 8 16 32 Samples per Target 325 350 375 400 425 450 475 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 20 Targets Algorithm 2D-DTSP 2D-DTSPN-ETRY 3D-DTSPN-E3D 3D-DTSPN-GWF 3D-DTSPN-RFAC 3D-METSPN-E3D 3D-METSPN-GWF 3D-METSPN-RFAC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 9 The line plots show the median non-dimensional tour cost of the different algorithms as the number of samples per target visibility volume increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' the algorithms that only consider one point per region have lower execution times than the algorithms that consider neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 2D ETRY method has a similar execution time to the 3D DTSPN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' However, the heuristic METSPN algorithm has a lower execution time compared to the other 3D methods because the graph that it creates is smaller and less computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The results suggest that for a large number of samples the METSPN algorithm outperforms the 3D DTSPN algorithms since it produces tours of similar cost but with a computation time that is approximately two orders of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Conclusion This paper studied the view planning problem of using a 3D Dubins airplane model to inspect points of interest in an urban environment in minimum time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Triangular meshes were used to compute approximate visibility volumes that correspond to locations where an unobstructed view of the target can be obtained while satisfying imaging and altitude constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The mesh-based approach for computing visibility volumes is flexible and can represent more complex geometries than have previously been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' A range-based sensor model was assumed here, however mesh-based view planning can potentially support other sensor models, sensing modalities, and encode sensing performance 15 100 150 200 250 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 5 Targets 250 300 350 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 10 Targets 200 225 250 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 15 Targets 325 350 375 400 Tour Cost (nondim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=') 20 Targets 5 10 15 20 Number of Targets 100 101 102 103 104 Time (s) Algorithm 2D-DTSP 2D-DTSPN-ETRY 3D-DTSPN-E3D 3D-DTSPN-GWF 3D-DTSPN-RFAC 3D-METSPN-E3D 3D-METSPN-GWF 3D-METSPN-RFAC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 10 The box plots show the range of cost across all sets of target visibility volumes when the number of samples per target visibility is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8 and the number of pitch angles is held at 𝑛𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The vertical black bars show the upper and lower quartiles of the data while the colored sections show the middle quartiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The black bar in the middle of the box plots is the median of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The black diamonds are outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The line graph shows the increase in computation time as the number of target visibility volumes increases on a log10 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The shaded region around each line shows the range of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' The 3D Dubins airplane model used in this work can, in some circumstances, produce more efficient inspection tours by exploiting altitude changes that are otherwise not possible with constant-altitude Dubins path tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' In cases where visibility volumes occupy disjoint altitude segments, the 3D algorithms provide a feasible solution where the 2D algorithms are not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' However, the pitch angle constraints of a Dubins airplane limit the change in altitude over a tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Altitude changes are accompanied by an increase in path length and thus are only efficient when they greatly improve access to the visibility volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This work introduced a heuristic that computes edge costs by replacing the 3D Dubins path computation with a simpler lower bound and assigning heading and pitch angles based on the geometric relation of successive points in a tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' This strategy provides a similar tour cost to other 3D algorithms that use the exact 3D Dubins path planner for edge cost computation but with computation time reduced by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Future work may consider the view planning problem in the presence of obstacles that must be avoided, with target visibility volumes that overlap, and/or with uncertain moving targets to be inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' Acknowledgments This work was supported by the William States Lee College of Engineering at the University of North Carolina at Charlotte through the Multidisciplinary Team Initiation (MTI) Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 16 References [1] Chitsaz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', and LaValle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=', “Time-optimal paths for a Dubins airplane,” 46th IEEE Conference on Decision and Control, IEEE, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' 2379–2384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE4T4oBgHgl3EQf3w3V/content/2301.05309v1.pdf'} +page_content=' https://doi.' 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0000000000000000000000000000000000000000..e71e9eef217dc0e2cba1a807245ba9d75bc9147e --- /dev/null +++ b/-dFJT4oBgHgl3EQfpiwj/content/tmp_files/2301.11600v1.pdf.txt @@ -0,0 +1,1821 @@ +Creative beyond TikToks: Investigating Adolescents’ Social +Privacy Management on TikTok +Nico Ebert +nico.ebert@zhaw.ch +Zurich University of Applied Sciences, +School of Management and Law +Winterthur, Zurich, Switzerland +Tim Geppert +Zurich University of Applied Sciences, +School of Management and Law +Winterthur, Zurich, Switzerland +Joanna Strycharz +University of Amsterdam, Faculty of +Social and Behavioural Sciences +Amsterdam, North Holland +Netherlands +Melanie Knieps +University of Zurich, Digital Society +Initiative +Zurich, Zurich, Switzerland +Michael Hönig +Zurich University of Applied Sciences, +School of Management and Law +Winterthur, Zurich, Switzerland +Elke Brucker-Kley +Zurich University of Applied Sciences, +School of Management and Law +Winterthur, Zurich, Switzerland +ABSTRACT +TikTok has been criticized for its low privacy standards, but lit- +tle is known about how its adolescent users protect their privacy. +Based on interviews with 54 adolescents in Switzerland, this study +provides a comprehensive understanding of young TikTok users’ +privacy management practices related to the creation of videos. +The data were explored using the COM-B model, an established +behavioral analysis framework adapted for sociotechnical privacy +research. Our overall findings are in line with previous research +on other social networks: adolescents are aware of privacy related +to their online social connections (social privacy) and perform +conscious privacy management. However, we also identified new +patterns related to the central role of algorithmic recommenda- +tions potentially relevant for other social networks. Adolescents +are aware that TikTok’s special algorithm, combined with the app’s +high prevalence among their peers, could easily put them in the spot- +light. Some adolescents also reduce TikTok, which was originally +conceived as a social network, to its extensive audio-visual capabil- +ities and share TikToks via more private channels (e.g., Snapchat) +to manage audiences and avoid identification by peers. Young users +also find other creative ways to protect their privacy such as identi- +fying stalkers or maintaining multiple user accounts with different +privacy settings to establish granular audience management. Based +on our findings, we propose various concrete measures to develop +interventions that protect the privacy of adolescents on TikTok. +KEYWORDS +TikTok, adolescent, video, privacy management, social privacy, +COM-B, Behavior Change Wheel +1 +INTRODUCTION +The global popularity and rapid growth of TikTok are accompanied +by problems that are the subject of public debate. The platform has +been criticized for several privacy issues such collecting personal +This work is licensed under the Creative Commons Attribu- +tion 4.0 International License. To view a copy of this license +visit https://creativecommons.org/licenses/by/4.0/ or send a +letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. +Proceedings on Privacy Enhancing Technologies 2023(2), 1–15 +© 2023 Copyright held by the owner/author(s). +https://doi.org/XXXXXXX.XXXXXXX +data from minors under the age of 13 [29, 33] or transferring U.S. +minor’s data to China [70]. These issues are especially alarming as +a large number of platform users are underage [74]. As a result, the +private space of children such as the bedrooms from which they +create their videos becomes visible to the world [43]. TikTok has +reacted to public criticism by introducing several features to better +protect the privacy of adolescents [75]. +Implicit to the current debate is the apparent consensus that +adolescents lack the awareness or skill set to consider the possible +privacy implications of their platform use. In fact, however, little +evidence is publicly available on how TikTok is used by adolescents +[63] and how they manage their privacy on the platform [40]. As +short videos are the platform’s key purpose, this immediately raises +the question of how younger users protect their privacy when they +create videos. In this paper, we aim to answer the following re- +search question: "How and why do adolescents manage their privacy +when creating videos on TikTok?" We build on the COM-B model for +behavioral analysis, an established conceptual framework for be- +havior change widely applied in health communication and beyond +[62]. This model allows us to explore not only privacy behavior, but +also related users’ motivations, skills and desires. To explore adoles- +cents’ privacy management in video creation from the perspective +of their capabilities, motivations, and opportunities, we conducted +interviews with 54 adolescent TikTok users in Switzerland, where +TikTok has gained popularity among young people [6]. +This study contributes to the growing body of research that ad- +dresses how adolescents manage their privacy on social networks. +This topic, which is often viewed (and judged) through the moral +lens of adults, is usually met with a sense of alarm. Empirical evi- +dence that could serve as a better fact base about adolescents’ online +privacy behavior is largely based on platforms that cater to a more +general population (e.g., Facebook, Twitter). However, TikTok is +not only explicitly geared towards a younger audience [47], but also +strongly encourages the sharing of short personal videos. Although +adding text is possible on TikTok, it takes a back seat in favor of +video content. Given the particularly sensitive nature of one’s image +and its link to an individual’s personal development [28], TikTok +strikes us as a particularly relevant but under-researched new use +case with the potential to enrich the ongoing debate about how – +if at all – teenagers perceive and manage their privacy [40]. +1 +arXiv:2301.11600v1 [cs.SI] 27 Jan 2023 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +This study is the first to examine how adolescents between the +ages of 12 and 18 manage their privacy on TikTok when it comes +to personal videos. Our findings are based on original data from +personal interviews and offer unique insights into how privacy +concerns influence young people’s online behavior. The qualitative +nature of our study helped us to understand the components that +shape sharing behavior on TikTok. Ultimately, this allowed us to +make concrete suggestions on how to effectively promote privacy- +protective behavior among adolescents on TikTok (e.g., specific +training, improved app features, and policy enforcement). +2 +RELATED WORK +2.1 +TikTok and Privacy Issues +As with many social media platforms, TikTok has come under +scrutiny for its handling of personal data. TikTok is a video-focused +social network originally started as U.S.-based musical.ly but later +bought by Beijing ByteDance Technology Ltd. The TikTok app +(available for Android and iOS) allows users to create short videos +(which may only be a few seconds long) and live streams [42]. Like +YouTube, TikTok is a manifestation of user-generated media where +content is not primarily created by a limited number of producers +but by a myriad of users [44]. Compared to other social networks +such as Facebook or Instagram, users on TikTok do not need to +communicate with each other to find a community. They can simply +visit the “For You” default page to find like-minded users [16, 42, 43, +87]. Via the primary button at the center of the home screen, users +can easily record and edit short videos, apply various effects and +sounds, and reuse content produced by other users. Videos can be +saved as drafts or published immediately to be viewed by different +audiences (myself, followers, everybody) [56]. As of September +2021, 1 billion monthly active users were reported [79], and 740 +million first-time installs were estimated in 2021 [73]. Cloudflare, a +provider of content delivery networks, ranked TikTok as the most +popular website of 2021, before Google [68]. TikTok is currently +also gaining in popularity among users below the official age limit +of 13 years [19]. In Switzerland, three-quarters of all adolescents +had a TikTok account in 2020 (behind Instagram and Snapchat with +both over 90%) [6]. Younger adolescents (12-15 years) were even +more likely to have a TikTok account than older adolescents (16-19 +years). Slightly more girls (78%) used it than boys (68%). 51% of all +adolescents stated to use it at least multiple times per week, and 38% +daily [6]. However, little is known about how young users think +about the data they share on the platform. +The app has raised numerous severe security and privacy con- +cerns (e.g., [5, 23, 24, 46, 84]) and caught the attention of the inter- +national authorities in the U.S. and EU [29, 69, 70]. For example, an +analysis of the app revealed extensive aggressive user tracking (e.g., +including techniques such as fingerprinting) and data sharing with +other websites (e.g., sharing searches with Facebook) [24]. The app +could also potentially collect other personal data from the user’s +smartphone (e.g., data from the clipboard [23]). Since young people +have always been an important user group of TikTok, concerns +have been raised about ByteDance’s handling of their personal +data. For example, in February 2019 ByteDance was fined USD +5.7 million by the U.S. Federal Trade Commission (FTC) because +musical.ly had collected information from minors under the age +of 13 in violation of the Children’s Online Privacy Protection Act +[33]. Due to the death of a 10-year-old TikTok user, the Italian data +protection authority has banned TikTok from processing the data +of users whose age could not be determined with full certainty [29]. +Also, the transfer of minors’ data to China after the acquisition +of U.S.-based musical.ly had caused a serious backlash in the US +and EU [69, 70]. As recent as June 2022, evidence surfaced that +ByteDance has repeatedly accessed U.S. user data from China – +a practice that they had denied three years earlier when similar +criticism was raised [26]. +TikTok has reacted to public criticism with several privacy- +related updates to the original app. As part of its settlement with +the FTC, the platform introduced an age-verification process for +its users based on self-declaration, meaning users can provide a +false age [48]. Further changes included extended parental control +features [41] and privacy settings contingent on the app users’ age +statement [75]. While children below 13 cannot use the app, ado- +lescents between the ages of 13 and 15 are automatically switched +to a “private account” as a default option, limiting those who can +view their videos to approved followers. When 16- and 17-year-old +users imitate an existing video in the form of a “duet” (split-screen +video) or “stitch” (video incorporating a short clip of someone else’s +content), these are automatically restricted to “friends only”. Only +users who are 18 and older can buy and send virtual gifts. However, +it is unclear if and how TikTok’s efforts have affected users’ privacy +management. +2.2 +Adolescents’ Privacy Management on +Social Media +From the moment adolescents started to use online social network- +ing sites, “online privacy” has been a major topic of discussion [31]. +Informational privacy can be defined as “the claim of individuals, +groups or institutions to determine for themselves when, how, and +to what extent information about them is communicated to others” +[81]. Research on online privacy and adolescents can be divided +into two categories: “institutional privacy” and "social privacy" +[67]. Institutional privacy refers to the data collection practices by +organizations (e.g., for commercial purposes) [67, 85]. The focus of +this paper is social privacy, i.e., issues related to sharing personal +information with others (e.g., friends and family). According to the +theory of "networked privacy," individuals do not have complete +control over the sharing of their personal information within social +connections (e.g., on social media) because privacy is not managed +by individuals alone, but by networks of individuals collectively +[58]. +Young people are often seen as particularly vulnerable social +media users with limited capacities to protect their privacy [15, 58]. +At the same time, they are also portrayed as individuals who put +themselves and others at risk with their naive and reckless social +media behavior [18]. Following this logic, numerous guides for +parents emphasize the importance of modifying privacy settings +and monitoring their children’s behavior (e.g., [37]). However, there +has also been a pushback to this alarmist perspective by scholars +who suggest that adolescents’ online privacy should be addressed +based on empirical research rather than paternal instinct [83]. +2 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +Empirical evidence from social networks other than TikTok (e.g., +Facebook) suggests that adolescents are aware of their social privacy +and actively manage their privacy on social media. As described +by boyd [9], adolescents want to avoid surveillance from parents, +teachers, friends and other meaningful persons in their lives (that +is what “online privacy” means to them). Adolescents’ social media +use seems to generally prompt increased disclosure of personal +information [72]. However, frequent sharing of content does not +imply that adolescents share indiscriminately, nor that the content +is intended for a wider audience [58]. Indeed, adolescents are con- +cerned about their privacy and capable of protecting it [1, 8, 17, 53]. +Contrary to conventional wisdom, young people are, in fact, more +likely to protect their privacy on social media than older people +[8]. Madden et. al found several strategies adolescents use on social +media to manage their identity and protect sensitive information +[57]. These strategies include deleting friends, faking names, delet- +ing content, withholding/faking information, and changing privacy +settings [20, 35, 64]. They also employed different “zones of privacy” +by using different channels for disclosing personal information to +maintain intimacy with friends while protecting their privacy from +their parents and strangers [50]. Privacy management can also +mean modifying social media content to shield it from audiences +[57, 64]. This practice is referred to as “social steganography” or en- +coding a message for a defined audience [58]. Adolescents’ privacy +management is influenced by various factors such as their social +environment (e.g., friends, parents), prior (negative) experiences as +well as the saliency of privacy settings [53, 86]. +Despite the existing evidence on adolescents’ social media use +on other social networks, researchers argue that existing findings +might not be directly applicable to TikTok [63]. Compared to other +networks such as Facebook or Instagram, TikTok mainly thrives +on content exploration and (re)-creation [87]. The focus is not on +the interaction between users and their social network but the +interaction with users’ videos proposed by an algorithm [7]. The +main feature, the “For You” page, presents an endless stream of +personalized, publicly available videos. Seeing them will motivate +users to react and create similar content (e.g., through features such +as “duet” or “stitch”). TikTok might therefore pose a particular threat +to adolescents’ privacy because a space previously conceptualized +as private and safe can easily become a space of public visibility, +surveillance, and judgment (such as in the case of a teenager being +seen to perform a dance routine in their bedroom) [43]. +Only a few studies have investigated adolescents’ privacy man- +agement on TikTok. There is some evidence that privacy manage- +ment on TikTok is considered as crucial by adolescents [16] and +becomes more stringent at higher perceived risks [40]. However, +it is unclear how and why adolescents manage their privacy on +TikTok. +2.3 +COM-B Model +As we were interested in the components that shape privacy be- +havior, we chose the COM-B model, which has been used in ex- +ploratory studies (e.g., [32]) and a series of contexts to change +behavior (e.g., [3]), as the conceptual framework for our analysis. +Many behavioral theories have been developed, often with overlap- +ping but differently named constructs [60] and limited guidance on +Capability +Motivation +Opportunity +Behavior +Reflective +Automatic +Psychological +Physical +Social +Physical +Environment +Individual +Figure 1: The COM-B model [62]. The three components capa- +bility (C), opportunity (O) and motivation (M) must be present for a +behavior (B) to occur. They interact over time and form a dynamic +system with positive and negative feedback loops [80]. +choosing an appropriate theory for a particular, real-world context +[62]. As a consequence, theories are often under-used to under- +stand real-world contexts and to design real-world solutions, which +makes replication, implementation, evaluation, and improvements +difficult [25, 62]. Researchers have argued that a comprehensive +meta-model or “supra-theory” model of behavior – like the COM-B +model – is needed that is applicable across contexts [25, 62]. As a +meta-model of behavior, the COM-B model does not come with a +pre-determined set of context-specific predictions that are common +for many behavioral theories. COM-B is based on several exist- +ing social cognition models and has a broader understanding of +behavior, having "also [...] automatic processing at its heart [like +emotions and habits], broadening the understanding of behaviour +beyond the more reflective, systematic cognitive processes that +have been the focus of much behavioural research [...] (for example, +social cognition models such as the Theory of Planned Behaviour)" +[62]. Its comprehensive nature and flexibility made it a good fit +for the exploratory nature of our study that was not constrained +by the conceptual boundaries of a single theoretical framework. +Furthermore, the model comes with hands-on actionable advice on +appropriate interventions in a given context in form of a holistic +behavior change framework (“Behavior Change Wheel”) (see [62]). +As illustrated in Figure 1, the COM-B model is based on three +components – capability (C), opportunity (O), and motivation (M) +– that shape a person’s behavior (B) [62]. Firstly, capability is a +subject’s psychological ability (including necessary comprehension, +knowledge, and skills) as well as the physical ability (e.g., control +of the body) to engage in a behavior. Secondly, motivation can +be defined as the subject’s mental processes that energize and di- +rect behavior. It includes the reflective motivation that involves +conscious processes (e.g., goals, plans, and evaluations) as well as +automatic processes (i.e., habitual, instinctive, drive-related, and +affective processes). Finally, opportunity is defined as an attribute +of the environmental system (unlike capability and motivation) +3 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +that enables or facilitates a behavior. Opportunities can be physical +(e.g., technical features of an app, material, financial, and time) and +social (e.g., norms and culture). In this study, we analyzed the par- +ticipants’ capabilities, opportunities, and motivation to engage in +privacy behaviors. +Firstly, capability is a subject’s psychological ability (including +necessary comprehension, knowledge, and skills) as well as the +physical ability (e.g., control of the body) to engage in a behavior. +Secondly, motivation can be defined as the subject’s mental pro- +cesses that energize and direct behavior. It includes the reflective +motivation that involves conscious processes (e.g., goals, plans, and +evaluations) as well as automatic processes (i.e., habitual, instinctive, +drive-related, and affective processes). Finally, opportunity is de- +fined as an attribute of the environmental system (unlike capability +and motivation) that enables or facilitates a behavior. Opportunities +can be physical (e.g., technical features of an app, material, financial, +and time) and social (e.g., norms and culture). +In our exploratory study, we did not focus on identifying inter- +actions between COM-B components that explain a specific target +behavior. Rather, our scope was first to learn about the full range +of behaviors and explanatory factors associated with adolescents’ +privacy management. +3 +METHODOLOGY +3.1 +Research Ethics +This paper is based on semi-structured, one-to-one interviews with +adolescents in the Canton of Zurich, Switzerland, conducted in +November 2021. In total, we visited two secondary schools (one +in the city of Zurich and one in the greater Zurich area) and three +youth centers (all in the city of Zurich). All interviews were audio- +recorded and transcribed verbatim. Ethical approval was obtained +from our university’s institutional review board. Study participants +provided written informed consent. For subjects below the age +of 16, additional consent was sought from the parents. The in- +terviews were voluntary and conducted at the institutions from +which the subjects had been recruited. Digital, personalized shop- +ping vouchers with a value of CHF 20 (~USD 19) were offered to +study participants as compensation. The amount and type of the +vouchers was determined beforehand together with the adolescents’ +supervisors (i.e., teachers, social workers) in order to not create an +inappropriate but still sufficient incentive. After the interviews, in +agreement with the participants, WhatsApp was used to deliver +the individualized vouchers to the participants and to allow them +to review their personal interview transcripts. +Several steps were taken to protect the participants identity with- +out compromising the transparency of our research process. To +begin with, all personally identifiable information was removed +(e.g., references to persons, locations) and participants’ names were +replaced with pseudonyms. Furthermore, the study data was stored +in line with our university’s storage policy and only the involved re- +searchers had access to the files. Finally, the original audio files were +deleted from all devices half a year after recording together with +other remaining personal data (e.g., phone numbers, WhatsApp +chats, digital vouchers). +3.2 +Sample and Procedure +Due to the lack of research on this topic, we chose a highly ex- +ploratory approach. To identify information-rich cases and make +optimal use of available resources, we drew a purposive sample +[27]. We used social media and search engines to find institutions +in the Canton of Zurich (e.g., secondary schools, youth work, youth +associations, museums) with contact with adolescents between 12 +and 18 years of age. Afterward, principals of participating insti- +tutions recruited interested teachers and social workers. They, in +turn, contacted interested TikTok users in the required age group. +To extend the participant base, we applied snowballing among the +interested TikTok users. Based on our primary aim (i.e., to explore +how adolescent TikTok users manage their privacy), we chose to +sample based on study participants’ age and gender (equally dis- +tributed). We decided to ignore other demographic information such +as ethnic identity. Following a pragmatic definition of theoretical +saturation [55], no new information emerged after approximately +40 interviews, and we ended data collection after the 54th interview. +We chose to employ semi-structured interviews for our study +because it encourages two-way communication and provides the +interviewer with the opportunity to learn the reasons behind an +answer. Some of the questions were part of the interviewer’s guide +(see Appendix), others were addressed at the moment. The inter- +view guide was developed based on a previous study that applied +the COM-B model in a qualitative setting [14] and adopted to the +context of the current study. After asking for demographic informa- +tion, we first explored general TikTok usage and motivation. The +other questions followed the COM-B structure and were related to +privacy-related behaviors as well as the explanatory components +related to the target behavior “video creation”. We finished the +interviews with questions about commercial privacy aspects (i.e., +targeted advertising and user tracking). After the interview pro- +cess was completed, study participants received a copy of their +interview transcript via WhatsApp and were invited to add infor- +mation or make amendments. Minimal revisions were made by one +participant. +To analyze the content of the interviews, we used a two-step pro- +cedure that first divided each statement into one of the four COM-B +components (behavior, capability, opportunity, motivation) before +further subdividing them into privacy-specific content. For phase +one, we used a directed content analysis approach [38] to analyze +the statements. To counter the subjectivity inherent to qualitative +data analysis, three researchers read and coded all statements into +the four COM-B domains (behavior, capability, opportunity, motiva- +tion). On the grounds of economy in both cost and effort, we decided +against using "intercoder reliability" (ICR). As full replication of +results was deemed unnecessary due to the exploratory and quali- +tative nature of data collection and analysis, we instead followed +guidelines suggesting the use of "multiple coding" which allows +independent researchers to cross check their coding strategies and +interpretation of data [2]. The authors engaged in researcher tri- +angulation [21] by discussing the emerging codes during the open +coding process of the first three interviews and developed cod- +ing guidelines. Disagreements were discussed and resolved. Using +the MAXQDA 2020 software, all responses were coded consistent +4 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +with six COM-B labels1 (behavior, psychological capability, auto- +matic/reflective motivation, social/physical opportunity). To ensure +continued adherence to the agreed coding guidelines, the three +researchers regularly communicated to ensure coding consistency. +In phase two, all statements – previously labeled as one of the +COM-B components – were further analyzed for their privacy- +specific content. Therefore, an inductive thematic analysis [11] +to identify themes within similarly coded statements was con- +ducted (see Appendix for coding scheme). One researcher identified +themes across identically coded statements and discussed them +with the other researchers. A theme reflects a collection of similar +responses from at least two different study participants. For exam- +ple, responses that were coded under the COM-B label “reflective +motivation” such as “I would be afraid of stupid remarks.”, “I have +no desire to be bullied.”, and “I can do without being ridiculed in +my class’s WhatsApp group.” were allocated to the privacy-specific +theme “negative reaction avoidance”. This step resulted in a list +of themes within each of the six COM-B labels. Ultimately, the +researchers reviewed and discussed the emerging themes, merged +similar themes, and re-labeled others. By playing the “devil’s ad- +vocate” – a common way to scrutinize identified themes [2] – we +sought to exploit the full potential of multiple coding to furnish +alternative interpretations of our findings. The anonymized, coded +interview transcripts are publicly available at osf.io/z8d3w. +4 +RESULTS +A total of 54 adolescents aged between 12-18 years (15 ± 1.82 years) +were interviewed, of which half (27) were female (see Table 1). +Interviews ranged from 5 to 21 minutes in length, with a mean of +12.6 min per interview (SD = 3.91). Most users attended secondary +school, and 80% had used the app for more than one year. Half +of the study participants admitted using TikTok between one and +three hours per day. +Table 1: Characteristics of one-to-one interview participants +(n = 54) +Variables +% (n) +Gender (% of females) +50% (27) +Age +15 ± 1.82 +Educational level +Primary level +2% (1) +Lower secondary level +54% (29) +Upper secondary level +44% (24) +User since +One year or less +20% (11) +Between one and two years +33% (18) +More than two years +46% (25) +Current app usage +Daily >= 3h +17% (9) +Daily >= 1 and <3h +50% (27) +Daily < 1h +28% (15) +Less than daily +6% (3) +1We did not need to code “physical capability” as the participants did not have physical +impairments. +Building on the conceptional framework of the COM-B model, +we identified 13 themes from the data analysis that described how +and why adolescents protect their privacy on TikTok (see Table 2). +These are described in more detail in the following. No weighting +was associated with the themes in terms of their overall contribu- +tion. +4.1 +Behavior +4.1.1 +Proactive privacy. The participants in our study mentioned +various ways to control the content of their TikToks2 and their +audience. Publishing content to audiences was described as reflec- +tive and non-automatic (as opposed to a habitual, non-reflective +publication of TikToks). This behavior is also referred to as the “ap- +proach” privacy strategy [58]. For example, regarding the content, +study participants described what they consider to be too sensitive +for publication on TikTok and would not publish (e.g., TikToks that +reveal too much about them). Lima (F, 14) creates public videos +and has 50 different accounts. She has clear privacy boundaries +regarding the video content: “I would not post TikToks where you +can see a lot of myself. I wouldn’t post videos in which I’m drunk.”. +Another form of restriction is to define who can see which type of +content on the platform. This includes TikTok users making drafts +only visible to themselves or blocking selected users from watching +videos. Bärbel (F, 13) actively tries to keep her parents from seeing +her videos: “To prevent my parents from seeing my videos, I can +simply block them.”. +We identified two subthemes within the proactive privacy theme: +private creators (19 persons, 35% of the sample) and public creators +(11 persons, 20%). Private creators create videos only for themselves +or close friends but do not publish them for a broad audience. A few +users described the practice of posting videos that are just visible to +themselves, only to be able to then repost them on “more private” +social media such as Snapchat or WhatsApp for a selected group +of people: “I don’t post my videos. I download them, save them +under photos, then send them on WhatsApp, for example. I only +use TikTok for editing.” (Amy, F, 17). +Public creators regularly create videos for their followers or the +general public. An extreme case is Joy (F, 13), who has used TikTok +since she was nine years old (when the app was still musical.ly). She +maintains 50 thematic user accounts with different age settings and +distinct followings (e.g., some accounts for gaming-related videos +and others for YouTube reposts). In addition to managing multiple +accounts, public creator Lima (F, 14) also uses the live feature. It +is available to users with at least 1,000 followers and allows them +to create personal live streams and interact with users in real time. +Lima had to set her age to 16 years to enable the live feature. +4.1.2 +Avoidance. Some study participants reported that they do +not publish videos on TikTok at all to protect their privacy. In the +literature, this is referred to as the avoidance privacy strategy [58]. +Peter (M, 14), one of 24 study participants (44%) we classified as a +pure consumer, stated: “I’ve never created a TikTok. I don’t even +know how to do it.”. Tim (M, 12) published once but decided to +only watch TikToks afterward: “To try it out, I uploaded something +2The term “TikToks” is used synonymously with videos. +5 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +Table 2: Identified themes for adolescents’ video privacy management on TikTok based on the COM-B model. Frequency is +calculated across 54 interviews. +Theme +Description +Frequency +Behavior +Proactive privacy +Publishing videos with control over the content and the audience +30 +Avoidance +Publishing no videos on the platform +24 +Capability (Psychological) +Past privacy incidents +Previous negative experiences related to privacy on the platform (e.g., lost +account, accidental publication) +15 +Privacy literacy +Knowledge and skills related to privacy management in the app (e.g., audience +understanding and configuration) +53 +Opportunity (Social) +Negative feedback +Negative behavior of others affects privacy management (e.g., observation of +cyber-bullying) +16 +Linkability experience +Observing that online personas can be linked to the personal sphere affects +privacy management (e.g., my teacher is on the platform) +39 +Restrictive influence +Restrictive behavior of others affects privacy management (e.g., restrictive +parental mediation) +34 +Opportunity (Physical) +Platform features +Privacy-related features of the platform (e.g., audience settings, sharing via +other social networks) +46 +Device features +Privacy-related features of the device (e.g., screen time limits, deleting videos +on the smartphone) +17 +Motivation (Automatic) +Negative emotion avoidance +Avoidance of negative emotions expected as a result of publication (e.g., shame, +fear) +15 +Motivation (Reflective) +Negative reaction avoidance +Goal to avoid expected negative consequences of publication +10 +Privacy identity +Privacy as a general value (e.g., also on other platforms) +5 +Publicity avoidance +Goal to avoid expected publicity of publication +29 +once, but nothing from me. I thought that was funny. But I prefer +to watch videos.” +4.2 +Capability (Psychological) +4.2.1 +Past privacy incidents. This theme refers to a specific form +of privacy-related knowledge (cp. [60]) gained after experiencing +potential or actual privacy incidents. Potential privacy incidents +are perceived as minor threats but may lead to increased privacy +awareness. “I posted my very first video by accident. It was only +seen by three people,” reported Yasmina (F, 15). Lima (F, 14), a public +creator, remembered: “I was half asleep and accidentally posted a +TikTok. The next morning, I saw that someone had commented +on the video. But I thought it was funny and not bad at all.” When +TikTok updated its app and increased the size of the “publish” button +to lower the threshold for publication, Lima decided to block app +updates. +Users have also realized that some of TikTok’s privacy features +can be easily bypassed. Their awareness of the platform’s weak- +nesses has contributed to a greater privacy awareness. An example +is a feature that allows blocking certain users from viewing videos, +which can be easily bypassed: “If I block people but they still want +to see my TikToks, they immediately make an extra fake account +and continue seeing them.” Roswitha (F, 15). However, she found +a way to manage her privacy: “Since these users have too few fol- +lowers, I simply block them again or ignore them depending on the +video.”. +A more serious subtheme are actual privacy incidents. Bärbel (F, +13) had to realize that she was not anonymizing herself sufficiently: +“I wore a mask on my face in the video, anonymously, so to speak. +But the people who deal with me every day recognized me by my +outfit, my room, and my hairstyle and posted the video in the class +WhatsApp chat.”. Anna (F, 14) reported losing her account and not +being able to reclaim it through TikTok’s customer support. At the +same time, other users were still able to watch her videos: “I made +videos of myself when I was 9 and then lost the account. Now the +videos are still public, but I can no longer access them.”. +4.2.2 +Privacy literacy. Privacy literacy can be defined as a com- +bination of factual or declarative (’knowing that’) and procedural +(’knowing how’) knowledge about online privacy [76]. Concern- +ing the publication of videos on the platform, adolescents need to +have the knowledge and skills to assess and manage audiences and +content as needed. +Respondents mentioned, for example, that the algorithm might +present a video on TikTok’s center stage: “It depends on how pop- +ular a video is and only then does it appear on the For You Page.” +(Bärbel (F, 13)) or that public videos can also be watched without +6 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +having a TikTok account: “From Google or Safari you can type in +TikTok and view the videos.” (Aron (M, 13)). They also described +how to find out which of their peers used TikTok: “When you post +a video, it spreads immediately and then you know who has TikTok +and who does not. Because so many people have TikTok now, it has +become weird for me to post TikToks.” (Elsa (F, 14)). Respondents +also described their audience and content management skills. The +private creator Bea (F,14) only publishes for a strictly curated list +of followers and therefore has established an approval process that +allows her to maintain the desired level of privacy: “I get to know +new classmates first and only then give them my TikTok account. +Afterward, they tell me they sent a request and I accept them as +followers in the app.” (Bea (F, 14)). Furthermore, the adolescents +interviewed were also able to assess different levels of sensitivity of +content in terms of their privacy and select an adequate audience +accordingly: “My buddy and I made 10 TikToks in which we share +our weekend activities with people. Some have 60,000 views. But +we think carefully what to make public.” (Alex (M, 18)). +The adolescents also talked about various app settings needed to +manage the audience, such as the activation of the private account +“Switching to the private account takes only two minutes. This is +not difficult.” (Alexandra (F, 12)) or knowing the publication status +of a video: “A draft is rendered greyish and blurry. When published, +it is bright and jumps right out at you.” (Alexander (M, 15)). Some +adolescents also perform “digital housekeeping” activities by re- +moving content related to a specific event or as a habit: “As I became +older, I started to delete old videos.” (Ariane (F, 15)). +4.3 +Opportunity (Social) +4.3.1 +Negative feedback. Negative feedback refers to expected or +observed negative feedback from others (such as harsh comments +to videos). Study participants reported negative reactions on the +platform (e.g., from strangers or people from the same school) as +an explanation for their privacy protection behavior. Alexander (M, +15) mentioned a general culture of mutual criticism: “Many of the +famous TikTokers sometimes make mistakes. Afterwards, everyone +makes fun of them in videos.”. Other respondents mentioned nega- +tive reactions from their peers that had influenced their behavior: +“A friend went viral with a video. Then she got yelled at on the +street. It would annoy me.” (Katja (F, 17)). +4.3.2 +Linkability experience. Similar to the perception of negative +feedback, the realization of how easily online personas can be linked +to the personal sphere can also lead to more restrictive publication +behavior. Study participants perceived the platform as a public space +shared by acquaintances and strangers. However, by recognizing +people from their school on their “For You” page, study participants +realized that they, too, could be easily recognized. As Georg (M, +15) put it: “There are maybe ten or twenty people in the school +building who do [public] TikToks regularly. You suddenly realize: I +know that guy from TikTok. That’s the reason why I don’t publish.”. +In addition to peers, respondents also described experiences that +made them understand that acquainted adults in authority positions +would be able to see their TikTok as well. Sibylle (F, 15) realized +this: “My music teacher was on TikTok singing a song.”. Therefore, +Sibylle also does not publish so as not to be recognized by everyone +on the platform. +4.3.3 +Restrictive influence. Restrictive influence refers to others +(e.g., close friends or parents) perceived to be restrictive or restrict- +ing study participants’ video creation behavior. Some interviewees +reported that their friends did not publish on TikTok, which in part +motivated why they did not publish, either. In mentioning his peers, +Felix (M, 12) stated: “Most of the people I know don’t upload any- +thing of themselves where they show their face.”. Another example +is restrictive mediation by parents or relatives: “My eight-year-old +cousin accidentally posted a video with my smartphone. His uncle +saw it on his For You page, so I deleted it.” (Sibylle (F, 15)). +4.4 +Opportunity (Physical) +4.4.1 +Platform features. Age verification is a key platform feature +intended to protect the privacy of young users (not limited to cre- +ating videos) and the subject of much public discussion. In the +semi-structured interviews, 29 of the interviewed participants were +also asked what age they provided. Two-thirds admitted that they +had given a false age when they registered (indicating, e.g., the age +of their parents). The main motivation for this behavior was to be +able to use TikTok in general (for those below the age of 13) or all +its features. Some study participants, like Martin (M, 14), also had +misconceptions about possible age restrictions: “Because otherwise, +TikTok won’t let me watch videos.”. +However, study participants also described how they used TikTok’s +features for privacy purposes in general. This includes using a nick- +name instead of their real name, limiting the use of personal infor- +mation on their profile page, and not linking their TikTok account +with other social media accounts (e.g., Instagram). While some in- +terviewees do not use a name at all: “Why should people know my +name? I have replaced my name and individual letters with an X.” +(Ali (M, 12)), others actively involve their parents to make use of +the in-app parental controls that restrict their app access. +Study participants also reported using various features related to +audience configuration, such as creating personal drafts, activating +a private account, deleting videos, or blocking users. Public creators +sometimes create multiple “privacy-tailored” user accounts with +specific follower groups for content of special sensitivity. Where the +features offered by the platform are perceived as too limited or in- +effective, the adolescents used creative workarounds not originally +anticipated by the platform provider. For example, it is not easily +possible to download and share drafts of videos that are not yet +published. Amy (F, 17), however, described a popular workaround: +“I post videos on TikTok, but only for me. Afterward, I’m able to +download them to share them with my friends on WhatsApp.” +4.4.2 +Device features. As part of the greater sociotechnical system, +some devices (e.g., smartphones) offer features that affect user pri- +vacy. For example, study participants make use of the “digital well- +being” functionality of their smartphone to limit their screentime: +“I used TikTok three hours a day because I didn’t know anything +better to do with myself. Now I’m trying to get a handle on this +with a screen time limit.” stated Matthias (M, 17). Sandra (F, 14) +was one of the study participants who used smartphone features to +share videos more selectively: "You can take a screenshot of drafts +with an iPhone and then send them via WhatsApp or Snapchat.". +As mentioned earlier, Lima (F, 14) noticed that the size of the red +“publish” button grew with each new app update compared to the +7 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +grey “save as draft” button. Fearing accidental publication, she by- +passed this potentially manipulative design pattern (“dark pattern”) +by using an old version of the app, which her operating system +allowed her to do: “Therefore, I have blocked the updates for TikTok +on my cell phone.”. +4.5 +Motivation (Automatic) +4.5.1 +Negative emotion avoidance. The interviewees describe vari- +ous negative emotions if they appeared in a video on TikTok. For +example, they mentioned feelings of discomfort, shame, awkward- +ness, and annoyance. Milo (M, 12), who does not publish any videos, +said: “I would be embarrassed to be seen in a video.” Elsa (F, 14) +reported that her desire to avoid negative emotions had evolved. +While she had posted videos on musical.ly, she didn’t publish on +TikTok anymore: “Posting TikToks has become weird for me.”. +4.6 +Motivation (Reflective) +4.6.1 +Negative reaction avoidance. Another reason for not pub- +lishing personal content was negative reactions by others to their +videos such as being bullied in class (e.g., in the WhatsApp class +chat). Alexander (M, 15), who does not publish any videos, com- +mented: “You make a mistake, people from school see it, it gets sent +on, and you get bullied.”. Avoidance can also relate to the negative +long-term consequences of sharing personal content. As they get +older, adolescents who are getting ready to join the job market real- +ize that their activity on TikTok could harm their career prospects. +“The Internet never forgets and if I eventually look for an appren- +ticeship, it may be that my future employer sees that. That’s very +bad for my reputation.” (Lima (F, 14)). +4.6.2 +Privacy identity. With privacy identity, we refer to a coher- +ent set of privacy-related behaviors and personal qualities of an +individual in a social setting [60]. Some teenagers consider privacy +as a value in itself and part of their identity. For example, for Yara +(F, 14), the publication of videos on TikTok is no different from +any social network activity: “It’s just not my thing. I don’t post in +general either, not even on Instagram or anything.”. Lena (F, 17) +explicitly stated that she considers privacy a significant personal +value: “Privacy is important to me. I keep everything private that +can be kept private.”. +4.6.3 +Publicity avoidance. Another motivation for restricting the +publication of personal videos on the platform is closely related to +the linkability experience theme: the desire to not attract public +attention. Study participants explained that publishing on TikTok +means being in the public eye: “It’s a big platform, and I don’t +want people around me to see that I make videos.” (Anna (F,14)). +While in musical.ly, the public was described as a community of +people with similar interests and ages, on TikTok, it is perceived +as a heterogenous, superficial place with different people of all +ages (including strangers, peers from the same school, teachers, +extended family members, and parents). Lina (F, 17) described how +the change in the audience had an impact on her behavior: “At +musical.ly, there were also strangers, but more my age. But TikTok +is now worldwide and there are adults everywhere. I don’t have +to post anything there.”. Her comment shows that the platform is +now perceived as completely public, whereas it used to be a more +private community. +5 +DISCUSSION +Our general observation of adolescents’ on TikTok is in line with +previous research on other social networks [1, 8, 17, 53]: Contrary +to public perception which portrays the publication of TikToks +by young people as automatic and unreflective, the adolescents in +our sample actively engaged in privacy management. They demon- +strated a strong awareness of the need to manage their online +identity and social privacy on the platform. However, the interview +participants were more concerned with protecting their privacy +from their immediate social environment than with institutional or +commercial privacy issues. That is, while they were generally aware +that TikTok used algorithms to tailor video content to their partic- +ular online behavior, they were more worried about the tangible +aspects of the algorithm: that a published video could immediately +appear on a classmate’s account. +Next, we will discuss the results in more detail following the +structure of the COM-B model. While many of our findings are +consistent with themes found in previous research on other social +media platforms (e.g., Facebook), a few themes and aspects are +indeed unique and – best to our knowledge – have not yet been +studied by researchers on TikTok or other platforms. The qualitative +nature of our data inform the design of very concrete interventions +on TikTok (Section 5.7). +5.1 +Behavior +In addition to previous research on other social networks [59], +we were able to identify two very different types of proactive pri- +vacy behavior: public and private creation. While public creators +perform privacy management to share videos directly on TikTok, +private creators merely use the platform to create and edit videos +to share them on other social networks that they see more appro- +priate for such content (e.g., Snapchat, WhatsApp). It indicates that +adolescents have different "imagined audiences" (mental conceptu- +alization of the people with whom the user is communicating, [49]) +on each social network and curate who sees what by switching +between networks. A unique finding of our study is that private +creators essentially reduce TikTok, which was originally conceived +as a social network, to its extensive audio-visual capabilities and +share their personal content where social connections already exist +and a higher degree of perceived control and intimacy exists (e.g., +WhatsApp). It is possible that such a practice might also be found +elsewhere (e.g., Instagram, YouTube). At a time when adolescents’ +increasingly use multiple social media platforms at once, privacy +perceptions of and management between different platforms has to +be addressed more comprehensively. That is, privacy management +can no longer be seen as a single-platform-phenomenon – an obser- +vation with important research implications. Rather than focusing +on isolated social networks with their own privacy standards, re- +searchers should expand their analysis to include a cross-network +view of privacy management. +8 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +5.2 +Psychological Capabilities +Similar to previous studies on other social media platforms [1, 52, +53], we found that adolescents possess knowledge and skills on how +to manage their privacy on TikTok (see "privacy literacy" theme). +That is, adolescents were not only able to assess the audience of +videos but also to actively manage the audience and content of +their TikToks. As previously noted [58], privacy management can +be very creative. This finding also holds true for TikTok: some +of our respondents reported using various accounts for different +audiences, blocking app updates to avoid receiving less privacy- +friendly versions of the app, and making an effort to detect fake +users trying to follow them. An interesting observation that can +potentially inform other research on social privacy management +in social networks is that adolescents on TikTok do not only use +the technical features provided by the social network itself. Instead, +some are also capable of using physical opportunities provided the +device (e.g., blocking app updates, screen time management). This +example illustrates how the existence of these generic physical +opportunities provided by the operating system can influence the +privacy management capability of young TikTok users to learn +about additional ways to protect their privacy. +In line with previous research we found that negative past expe- +riences affect future privacy management behaviors [53]. Incidents +can even serve as a learning opportunity [82]. In our sample, partici- +pants experienced near or actual privacy incidents (e.g., accidentally +publishing videos, loss of account with personal videos) that led +them to adapt their privacy management (e.g., immediately deleting +accidentally published videos, paying more attention to a publi- +cation in the future). While our data support the hypothesis that +incidents serve as learning opportunities, it must be said that cer- +tain very extreme violations of privacy (e.g., persistent bullying or +stalking) have not been reported in our study. It is unclear how such +experiences affect privacy behavior in the long run. Nonetheless, +our findings inform future research by showing that even minor pri- +vacy incidents without severe consequences can lead to improved +capabilities. +5.3 +Physical Opportunities +Adolescents in our sample used various features of TikTok and +the operating system to manage their privacy (themes platform +features and device features). At the same time, they were aware +of TikTok’s privacy management limitations (e.g., the ineffective- +ness of blocking users). Some of the measures TikTok has taken to +protect the privacy of younger users in response to public criticism +may not be very effective. Out of 29 study participants with whom +we discussed the topic, two-thirds used a false age. Many teenagers +we interviewed have been publishing on TikTok much before the +legally allowed age of 13. Regardless of the normative standpoint, +this calls into question TikTok’s fine-grained, age-based privacy +features. Despite legislative measures such as the Children’s Online +Privacy Protection Act of 1998, this problem has been described on +other social networks in the past [51, 54]. Sometimes also parents +help their underage children to access social networks [10]. Rea- +sons for using social networks below the specified minimum age +are diverse (e.g., wanting to stay in touch with classmates, want- +ing unrestricted access to TikTok’s features) [10]. Consequently, +technical measures to protect children such as non-public accounts +or content restrictions are failing [65]. boyd et. al [10] called for +abandoning ineffective age-based mechanisms. Instead, she advo- +cates for an honest discussion about children’s use of social media +and a rethinking of the industry to better incorporate the needs of +children and parents when developing apps. +Another issue on social networking sites is account loss [66]. +This issue was also highlighted by several of our respondents who +reported that they were unable to reclaim a video they had posted +after losing an account. As a consequence, they were unable to +revoke their consent from publishing a childhood experiment that +would now remain online forever. This is particularly problematic +against the background of increasingly better algorithms for rec- +ognizing people in images and videos and the resulting linkability +risk (e.g., Clearview AI [36]). To exercise the "right to be forgotten" +as embodied in the EU GDPR, for example, the ability to reclaim +accounts and delete old videos is essential. It is unclear whether ac- +count loss among adolescents is a broader phenomenon or whether +other social networks are affected as well. +5.4 +Social Opportunities +Our findings on TikTok support previous research demonstrating +that the social environment of teenagers shapes their privacy be- +haviors [53]. Other social network users as well as the parents are +major agents of socialization [31]. Social norms, which emerge as +a response to observed behavior or expected attitudes of friends +and parents, influence children’s intention to share personal infor- +mation [77]. If friends and parents disapprove of such behavior, +children tend to share less. A recent study on TikTok described, that +restrictive mediation by parents can also lead to more restrictive +disclosure behavior in children [40]. +In our study, we identified similar social influences on TikTok. +Observing strangers being publicly criticized for videos (theme +negative feedback) resulted in restrictive publication behavior by +the adolescents we interviewed. In line with previous research [77], +the restrictive norms and behavior of relatives, parents, and friends +were also found to have the potential to affect behavior on TikTok +(e.g., not publishing or blocking parents from videos). +What makes TikTok stand out from other social networks, is its +specific content algorithm based on a granular observation of user +preferences [45]. The results of our study indicate that prevalent +TikTok usage among peers in combination with the platform’s spe- +cific algorithm that immediately displays the published content to +cohorts with similar attributes – i.e., peers – may increase the social +influence of others on adolescents’ privacy behavior (“linkability +experience”). Unlike posting a video under a nickname on YouTube +that may never be discovered by peers, adolescents were aware that +posting on TikTok was potentially more privacy-invasive. They +recognized that their videos could become visible to their personal +environment (e.g., in the schoolyard). This experience led to re- +stricted publication behavior. +5.5 +Automatic and Reflective Motivations +Adolescents’ motivations for protecting their privacy on TikTok +were based on either wanting to avoid publicity, to avoid nega- +tive reactions/emotions, or to actively achieve privacy (themes +9 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +negative reaction avoidance, publicity avoidance). The adolescents +interviewed reported wanting to evade the public eye and feared +negative feedback (e.g., public criticism). These are themes previ- +ously described on other social networks [53]. To avoid a negative +emotional outcome (e.g., shame), they refrain from having a too +public profile (theme negative emotion avoidance) (see [13] for a +similar finding). +For some adolescents, privacy was a personal matter beyond +TikTok (theme privacy identity). That is, these teenagers were in- +trinsically motivated to keep their information private - a finding +that stands in contrast with previous research on other social net- +works. Research suggests that, on average, adolescents have fewer +privacy concerns than young adults [4, 22]. However, our findings +indicate that these concerns can vary greatly across adolescents, +and some may place great value on their privacy on social media. +Even though the theme was mentioned by only a few participants, +it underscores that adolescents are not a homogeneous group when +it comes to motives for managing privacy on social media. For some +participants’ being private is a personal value and their goal is to +achieve a coherent privacy behavior on TikTok and beyond. +5.6 +Methodological Consideration +For our study, the COM-B model helped to holistically understand +adolescents’ privacy management on TikTok related to the creation +of videos. It has a solid theoretical foundation and – according to its +authors – can be applied across various contexts. However, much of +the research to date has applied the COM-B model to health-related +behaviors such as smoking cessation and lowering cardiovascular +disease risk [60]. Our study, which showed that the COM-B model is +also a suitable analytical framework for studying privacy behavior, +provides yet another use case. By demonstrating its relevance to +the privacy management of adolescents, we strengthen the model’s +extrinsic validity. +5.7 +Possible Approaches for Privacy +Interventions +Several of the themes we identified can be used as starting points +for the development of privacy interventions. The COM-B model is +part of a theory-driven intervention development framework called +behavior change wheel (BCW), a synthesis of behavior change +frameworks [62]. In the logic of the BCW, interventions are di- +rected at desired “target behaviors” (e.g., enabling privacy settings). +Building on the interview findings and our observations, Figure 2 +shows different parties and ideas for potential target behaviors af- +fecting adolescents’ video privacy management. It focuses on which +behaviors to address and does not answer the question of how to +design interventions that address these behaviors (e.g., adequate +behavior change techniques [61]). +Any intervention schemes to improve the privacy of adoles- +cent TikTok users should focus on the behavior of the adolescents +themselves. The interviews provide concrete suggestions for be- +haviors that adolescents already report that improve their privacy +protection. This includes encouraging young users to remove in- +appropriate videos from the platform and the use of alternative +social media apps (e.g., WhatsApp) to share content (theme: proac- +tive privacy). Some of our participants reported regular checks if a +video with the status “published” should be set to private. They also +removed their old TikToks from the app and their smartphone. Our +private creators did seldom publish on TikTok but used alternative +apps such as Snapchat or WhatsApp with a perceived higher level +of privacy and the ability to automatically delete shared TikToks +after being watched by their friends. Another possible target behav- +ior derived from our observations is “backing up user credentials” +(theme: privacy literacy). Some adolescents in our sample who had +already created accounts in musical.ly could not delete published +videos because they had forgotten their credentials, and were not +able to prove their identity to the TikTok support to retrieve their +account. An intervention could mitigate account loss, especially +in cases where children have multiple accounts. Finally, teenagers +should be made aware of the privacy settings (e.g., the private ac- +count) and the potential risks of not correctly setting these (theme: +reflective motivation). For example, in our interviews, participants +accidentally published a TikTok upon their first usage of the app +because they were not aware others would immediately see it. +The platform must also play an important role in safeguarding +the privacy of children and adolescents. Improving features directly +related to privacy such as improved age verification, more effec- +tive blocking of users, and facilitating access to lost user accounts +are promising approaches (theme: platform features). As described +earlier, many adolescents in our sample did not use their real age +due to various reasons. For example, they were often unaware that +the private account would have been activated by default if they +had provided their real age. As a result of providing false infor- +mation, the privacy settings were much more lenient and TikTok +videos would not only be published to followers but to everybody. +Following boyd et. al’s [10] philosophy, one possibility would be +to abandon TikTok’s age-based mechanism and incorporate the +needs of children and parents when developing the app. For TikTok, +this could mean taking a certain level of responsibility for its con- +tent and giving kids and parents ways to control what videos are +shown (e.g., via a content configuration or a separate app similar +to YouTube Kids). Even if the app adhered to the age-based privacy +concept, describing the consequences of providing real age (e.g., +better privacy protection) might encourage some youth to provide +their real age. Another approach has been lately launched by the +twin app Douyin [30]. Douyin introduced an age verification that +is not based on self-declaration only but requires – unlike the in- +ternational counterpart TikTok - user authentication and imposes +restrictions on the permitted daily use for users under 14. +Some participants also criticized that they could not effectively +block users who they wanted to prevent from seeing their videos. +The problem persists because blocked users can immediately "respawn" +under a different username. TikTok could prevent this issue with +a feature that block all accounts of the same user (similar to Insta- +gram [39]). Some study participants also reported feeling “nudged” +by the user interface design towards publishing TikTok video for a +broad audience. Others described publishing personal TikToks acci- +dentally. While nudging teenagers towards better privacy behavior +is also controversial [78], presenting them with simple alternatives +(such as publishing a TikTok vs saving a local draft) could provide a +welcome middle ground. Furthermore, TikTok might also do more +to educate its users on how to protect their privacy. This suggestion +10 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +TikTok +Users +Family & +Friends +Schools +& Youth +Work +Policy- +Makers & +Privacy +Advocates +Other +Platform +Users +OS +Vendors & +Other Apps +ByteDance +(TikTok +Creator) +Enable +privacy +settings +Share/remove +personal content +consciously +Backup +account +credentials +Share TikToks +using other apps +(e.g., Whatsapp) +Inform each other +about privacy +possiblities +Do not nudge +users towards +publication +Educate children about long- +term privacy risks +Improve privacy +features (e.g., +reclaiming ‘lost’ +accounts, age +verification) +Use TikTok yourself to +understand privacy +issues +Educate students early +about longterm +privacy risks +Support childrens’ +privacy efforts +Provide privacy +tutorials +Create & enforce +privacy laws +(e.g., transparency +about PII usage, +age verification) +Housekeeping +functionality for +TikTok +Enforce app +privacy in OS +Explain business +model of TikTok +Use TikTok yourself to +understand privacy +issues +Explain business +model of TikTok +Respect the privacy +of others +(e.g., norms) +Figure 2: Different parties and their potential target behaviors relevant for adolescents’ video privacy management on TikTok +is based on our observation that capabilities varied between adoles- +cents and TikTok users had begun to create such privacy tutorials. +The latter indicates a demand for more support (e.g., via privacy +tutorials provided by TikTok). +Family, friends, schools, and youth workers can also positively in- +fluence the privacy management of adolescents (social opportunity +themes). In addition to supporting adolescents’ privacy efforts, their +social network could use TikTok themselves to better understand +specific privacy issues. In our sample, an uncle of an eight-year-old +boy used TikTok himself and warned him about the possibility on +TikTok of publishing a video by accident. The social environment +can also advise about long-term privacy risks to the children and +adolescents of which they might not yet be aware. Among a group +of adolescents of the same class, we repeatedly heard the narrative +of a classmate being recognized on TikTok despite her wearing +a mask. Due to this “risk narrative” the whole class was aware +of the potential risks of insufficient anonymization on TikTok. A +collection of such tales could be used by teachers in the classroom +to illustrate the privacy risk associated with the platform. +As users do not only interact with each other when they share +videos but also with the platform and its owner company, teenagers +should also be made aware of commercial privacy issues. Our data +confirmed that adolescents’ primary privacy focus was indeed so- +cial. To this end, adolescents would need to understand TikTok’s +business model, which heavily relies on their personal data, and +the organization behind TikTok. +Policymakers and privacy advocates are also relevant actors. Not +only do they seek to create privacy laws to protect users but also to +enforce these laws through, for example, insisting on effective age +verification (theme: platform features). Ideally, these actions are +guided by evidence in collaboration with researchers, adolescent +users, and parents. For example, our findings indicate that ado- +lescents did not know that TikTok had taken additional measures +to protect them in 2021 [75]. While privacy legislation demands +transparency for data subjects – especially for children – this ex- +ample shows that there is room for improvement in terms of the +implementation of laws. +It should also be mentioned that other TikTok users can influence +an adolescent’s privacy behavior (social opportunity themes). Older +and more experienced teenagers may have capabilities (e.g., based +on their negative experiences) that can benefit younger and less +experienced users. One of our participants reported having learned +about privacy settings from a video on TikTok. Indeed, some more +experienced users have already begun to acts as mentors. This +includes the user @seansvv with 1.1 million followers, who stated +in his biography “I Read ToS [Terms of Service] So That You Don’t +Have To” and regularly posts TikTok videos related to privacy topics +[71]. +Finally, our interviews showed that OS vendors and the vendors +of other apps contribute to teenagers’ privacy on TikTok (theme: +device features). OS vendors have implemented more and more +privacy control mechanisms for their end-users (e.g., granular rights +management, location sharing notifications). These methods all +work on low-level personal data (e.g., IP address, location, and +email address). However, videos shared by adolescents on TikTok +that possibly contain more sensitive personal data with higher risks +involved are not yet covered by these mechanisms. At times when a +user publishes a video accidentally, the OS could warn them in the +same way that they are warned when sharing their location with +the app. In our sample, participants reported also manually cleaning +11 + +Proceedings on Privacy Enhancing Technologies 2023(2) +Ebert et al. +up their TikToks in the app and on their phones. OS vendors could +provide housekeeping functionalities that would simplify removing +personal content across different social networks and on the phone. +5.8 +Limitations and Future Research +As with most qualitative research, our sample is small and was not +drawn randomly. Therefore, we cannot claim that the results are +representative of all young people in the region under consideration, +and certainly not of Switzerland as a whole. Further validation +with different samples is needed to strengthen the findings (e.g., +including subjects’ socioeconomic status). +Choosing interviews as our data collection methodology was use- +ful to learn more about the perspectives of adolescents in Switzer- +land. Nevertheless, we are aware of the limitations associated with +this method. Primarily, we relied on self-reporting rather than be- +havioral observations. Self-reports can be biased due to various +influences, such as subjects’ desire to portray themselves in a posi- +tive light. Future studies might want to gather data from a wider +range of sources, such as direct observations of privacy manage- +ment behavior (e.g., through TikTok data donations). +Based on our findings, future research could develop and system- +atically test privacy interventions based on the BCW methodology. +A necessary first step would be to identify appropriate target be- +haviors with the greatest potential to improve privacy management +among adolescents. Our research could be a starting point for select +a “promising” target behavior reported by the adolescents (e.g., ac- +tivating the private account) to address in a target population (e.g., +pupils of a local school). To identify a baseline for each of the poten- +tial behaviors and to select a target behavior among them, further +research would be necessary (e.g., in form of a survey among pupils). +Furthermore, additional research is required to select appropriate +behavior change techniques (e.g., increasing awareness for privacy +settings) and evaluate their effectiveness (e.g., with an experiment). +Importantly, such research could also control for factors such as +socioeconomic status might also be relevant to explain privacy- +related behaviors on TikTok [34]. Given that teenagers may have +very heterogeneous privacy management capabilities, motivations, +and opportunities, depending on their age and experience regarding +the platform, interventions need to be tailored to the specific target +group. Large-scale intervention studies using the BCW can help to +identify effective and evidence-based policies to improve privacy +management among young people on social media platforms like +TikTok. +Our interviews focused on social aspects of adolescents’ privacy +management. That is, our interviewees were more concerned with +protecting their privacy from their social environment than from +the corporations dealing with their data for commercial purposes; +see [53]. Yet, TikTok videos are not only shared with other users +but also with ByteDance. Even the users we identified as pure +consumers who only view but not create content may have privacy +issues. As the video and ad algorithms are known for their high +level of customization, they make the platform heavily reliant on +personal data including detailed user behavior [45]. Both users’ +active and passive behavior on the app has consequences: The +TikTok pixel allows companies to engage in detailed web tracking +of TikTok users on websites (e.g., a user who sees the ad on TikTok +might buy the product in the online shop) [12]. Further research +could investigate if adolescent users are aware of these commercial +privacy aspects and how they manage them. +ACKNOWLEDGMENTS +The research reported in this article was funded by the Digital +Future Fund (DFF), which is part of the Digitalization Initiative of +the Zurich Higher Education Institutions (DIZH), Switzerland. We +would like to thank all adolescents, teachers, and social workers +we contacted in conducting our study. We also thank Frank Wieber, +Katja Kurz and Manuel Günther for their helpful comments. +REFERENCES +[1] Claire Balleys and Sami Coll. 2017. Being Publicly Intimate: Teenagers Managing +Online Privacy. 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When did you start to use TikTok? +(3) Do you remember how old you were when you started using +the app? +(4) How many people do you follow? How many followers do +you have? +(5) Do you share videos? How many? What types of videos? +(Behavior) +(a) Why do you/don’t you share videos? (Motivation) +(b) If yes: How do you share videos on TikTok? (Psychological +capability) +(6) Who can see your videos when they are shared? (Psycholog- +ical capability) +(7) How can you influence who can see your videos? (Psycho- +logical capability) +(8) Do you restrict who can see your videos? (Behavior) +(a) If yes: Why / When do you restrict your videos? (Motiva- +tion) +(b) If no: Why? Have you ever considered restricting your +videos? (Motivation) +(9) Do your friends or others restrict their/your videos? (Social +Opportunity) +(10) Have you ever accidentally posted a video? If yes: What did +you do? (Psychological capability) +(11) What do you think about TikTok’s features to share/restrict +videos? (Physical opportunity) +A.2 +Coding Scheme +Table 3 shows the hierarchical coding scheme together with the +frequency of each code calculated across the 54 interviews. +14 + +Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok +Proceedings on Privacy Enhancing Technologies 2023(2) +Table 3: Coding Scheme (translated from German). Frequency is calculated across 54 interviews. +Code +Description +Frequency +Usage_since +Start of TikTok usage +54 +Usage_frequency +Frequency of TikTok usage +53 +App_age +Age entered into the app at first use +29 +Video_Behavior +Avoidance +I normally do not create/publish TikToks. +24 +Proactive +PersonalCreator +I regularly create/publish TikToks for myself and close friends. +19 +PublicCreator +I regularly create/publish TikToks for my followers/the public. +11 +Video_PsyCapability +PastPrivacyIncidents +Minor +I have perceived a potential/minor privacy incident. +9 +Severe +I have perceived a severe privacy incident. +8 +PrivacyLiteracy +AudienceContentLiteracy +I’m aware of different audience/content types and have the ability to manage +them. +52 +TechnicalLiteracy +I have the technical knowledge and skills to manage my audience. +50 +Video_SocOpportunity +NegativeFeedback +Others show negative reactions to TikToks, that’s why I’m not active. +16 +LinkabilityExperience +Users can be easily recognized in real life. +39 +RestrictiveInfluence +I’m not active because others are also restrictive or enforce my privacy. +34 +Video_PhyOpportunity +PlatformFeatures +TikTok helps to ensure my privacy. +46 +DeviceFeatures +The device helps to ensure my privacy. +17 +Video_AutMotivation +NegativeEmotionAvoidance +I don’t publish content to avoid negative emotions. +15 +Video_RefMotivation +NegativeReactionAvoidance +I don’t publish content to avoid negative reactions. +10 +PrivacyIdentity +I don’t publish content because privacy is important to me. +5 +PublicityAvoidance +I don’t publish content because I don’t want to be in the public eye. +29 +15 + diff --git a/-dFJT4oBgHgl3EQfpiwj/content/tmp_files/load_file.txt b/-dFJT4oBgHgl3EQfpiwj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..483c7fd07e8564bf5de71a279c39a245d1090d9b --- /dev/null +++ b/-dFJT4oBgHgl3EQfpiwj/content/tmp_files/load_file.txt @@ -0,0 +1,1415 @@ 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+page_content=' Switzerland Tim Geppert Zurich University of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' School of Management and Law Winterthur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Switzerland Joanna Strycharz University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Faculty of Social and Behavioural Sciences Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' North Holland Netherlands Melanie Knieps University of Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Digital Society Initiative Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Switzerland Michael Hönig Zurich University of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' School of Management and Law Winterthur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Switzerland Elke Brucker-Kley Zurich University of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' School of Management and Law Winterthur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Switzerland ABSTRACT TikTok has been criticized for its low privacy standards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' but lit- tle is known about how its adolescent users protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Based on interviews with 54 adolescents in Switzerland, this study provides a comprehensive understanding of young TikTok users’ privacy management practices related to the creation of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The data were explored using the COM-B model, an established behavioral analysis framework adapted for sociotechnical privacy research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our overall findings are in line with previous research on other social networks: adolescents are aware of privacy related to their online social connections (social privacy) and perform conscious privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, we also identified new patterns related to the central role of algorithmic recommenda- tions potentially relevant for other social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents are aware that TikTok’s special algorithm, combined with the app’s high prevalence among their peers, could easily put them in the spot- light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some adolescents also reduce TikTok, which was originally conceived as a social network, to its extensive audio-visual capabil- ities and share TikToks via more private channels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Snapchat) to manage audiences and avoid identification by peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Young users also find other creative ways to protect their privacy such as identi- fying stalkers or maintaining multiple user accounts with different privacy settings to establish granular audience management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Based on our findings, we propose various concrete measures to develop interventions that protect the privacy of adolescents on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' KEYWORDS TikTok, adolescent, video, privacy management, social privacy, COM-B, Behavior Change Wheel 1 INTRODUCTION The global popularity and rapid growth of TikTok are accompanied by problems that are the subject of public debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The platform has been criticized for several privacy issues such collecting personal This work is licensed under the Creative Commons Attribu- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To view a copy of this license visit https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Proceedings on Privacy Enhancing Technologies 2023(2), 1–15 © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='XXXXXXX data from minors under the age of 13 [29, 33] or transferring U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' minor’s data to China [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' These issues are especially alarming as a large number of platform users are underage [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As a result, the private space of children such as the bedrooms from which they create their videos becomes visible to the world [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok has reacted to public criticism by introducing several features to better protect the privacy of adolescents [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Implicit to the current debate is the apparent consensus that adolescents lack the awareness or skill set to consider the possible privacy implications of their platform use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In fact, however, little evidence is publicly available on how TikTok is used by adolescents [63] and how they manage their privacy on the platform [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As short videos are the platform’s key purpose, this immediately raises the question of how younger users protect their privacy when they create videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In this paper, we aim to answer the following re- search question: "How and why do adolescents manage their privacy when creating videos on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='" We build on the COM-B model for behavioral analysis, an established conceptual framework for be- havior change widely applied in health communication and beyond [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This model allows us to explore not only privacy behavior, but also related users’ motivations, skills and desires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To explore adoles- cents’ privacy management in video creation from the perspective of their capabilities, motivations, and opportunities, we conducted interviews with 54 adolescent TikTok users in Switzerland, where TikTok has gained popularity among young people [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This study contributes to the growing body of research that ad- dresses how adolescents manage their privacy on social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This topic, which is often viewed (and judged) through the moral lens of adults, is usually met with a sense of alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Empirical evi- dence that could serve as a better fact base about adolescents’ online privacy behavior is largely based on platforms that cater to a more general population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Facebook, Twitter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, TikTok is not only explicitly geared towards a younger audience [47], but also strongly encourages the sharing of short personal videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Although adding text is possible on TikTok, it takes a back seat in favor of video content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Given the particularly sensitive nature of one’s image and its link to an individual’s personal development [28], TikTok strikes us as a particularly relevant but under-researched new use case with the potential to enrich the ongoing debate about how – if at all – teenagers perceive and manage their privacy [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='11600v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='SI] 27 Jan 2023 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This study is the first to examine how adolescents between the ages of 12 and 18 manage their privacy on TikTok when it comes to personal videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our findings are based on original data from personal interviews and offer unique insights into how privacy concerns influence young people’s online behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The qualitative nature of our study helped us to understand the components that shape sharing behavior on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ultimately, this allowed us to make concrete suggestions on how to effectively promote privacy- protective behavior among adolescents on TikTok (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', specific training, improved app features, and policy enforcement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 TikTok and Privacy Issues As with many social media platforms, TikTok has come under scrutiny for its handling of personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok is a video-focused social network originally started as U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='-based musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly but later bought by Beijing ByteDance Technology Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The TikTok app (available for Android and iOS) allows users to create short videos (which may only be a few seconds long) and live streams [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Like YouTube, TikTok is a manifestation of user-generated media where content is not primarily created by a limited number of producers but by a myriad of users [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Compared to other social networks such as Facebook or Instagram, users on TikTok do not need to communicate with each other to find a community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They can simply visit the “For You” default page to find like-minded users [16, 42, 43, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Via the primary button at the center of the home screen, users can easily record and edit short videos, apply various effects and sounds, and reuse content produced by other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Videos can be saved as drafts or published immediately to be viewed by different audiences (myself, followers, everybody) [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As of September 2021, 1 billion monthly active users were reported [79], and 740 million first-time installs were estimated in 2021 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Cloudflare, a provider of content delivery networks, ranked TikTok as the most popular website of 2021, before Google [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok is currently also gaining in popularity among users below the official age limit of 13 years [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In Switzerland, three-quarters of all adolescents had a TikTok account in 2020 (behind Instagram and Snapchat with both over 90%) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Younger adolescents (12-15 years) were even more likely to have a TikTok account than older adolescents (16-19 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Slightly more girls (78%) used it than boys (68%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 51% of all adolescents stated to use it at least multiple times per week, and 38% daily [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, little is known about how young users think about the data they share on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The app has raised numerous severe security and privacy con- cerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', [5, 23, 24, 46, 84]) and caught the attention of the inter- national authorities in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' and EU [29, 69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, an analysis of the app revealed extensive aggressive user tracking (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', including techniques such as fingerprinting) and data sharing with other websites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', sharing searches with Facebook) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The app could also potentially collect other personal data from the user’s smartphone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', data from the clipboard [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Since young people have always been an important user group of TikTok, concerns have been raised about ByteDance’s handling of their personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, in February 2019 ByteDance was fined USD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='7 million by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Federal Trade Commission (FTC) because musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly had collected information from minors under the age of 13 in violation of the Children’s Online Privacy Protection Act [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Due to the death of a 10-year-old TikTok user, the Italian data protection authority has banned TikTok from processing the data of users whose age could not be determined with full certainty [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Also, the transfer of minors’ data to China after the acquisition of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='-based musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly had caused a serious backlash in the US and EU [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As recent as June 2022, evidence surfaced that ByteDance has repeatedly accessed U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' user data from China – a practice that they had denied three years earlier when similar criticism was raised [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok has reacted to public criticism with several privacy- related updates to the original app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As part of its settlement with the FTC, the platform introduced an age-verification process for its users based on self-declaration, meaning users can provide a false age [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Further changes included extended parental control features [41] and privacy settings contingent on the app users’ age statement [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While children below 13 cannot use the app, ado- lescents between the ages of 13 and 15 are automatically switched to a “private account” as a default option, limiting those who can view their videos to approved followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' When 16- and 17-year-old users imitate an existing video in the form of a “duet” (split-screen video) or “stitch” (video incorporating a short clip of someone else’s content), these are automatically restricted to “friends only”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Only users who are 18 and older can buy and send virtual gifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, it is unclear if and how TikTok’s efforts have affected users’ privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Adolescents’ Privacy Management on Social Media From the moment adolescents started to use online social network- ing sites, “online privacy” has been a major topic of discussion [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Informational privacy can be defined as “the claim of individuals, groups or institutions to determine for themselves when, how, and to what extent information about them is communicated to others” [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Research on online privacy and adolescents can be divided into two categories: “institutional privacy” and "social privacy" [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Institutional privacy refers to the data collection practices by organizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', for commercial purposes) [67, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The focus of this paper is social privacy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', issues related to sharing personal information with others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', friends and family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' According to the theory of "networked privacy," individuals do not have complete control over the sharing of their personal information within social connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', on social media) because privacy is not managed by individuals alone, but by networks of individuals collectively [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Young people are often seen as particularly vulnerable social media users with limited capacities to protect their privacy [15, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' At the same time, they are also portrayed as individuals who put themselves and others at risk with their naive and reckless social media behavior [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Following this logic, numerous guides for parents emphasize the importance of modifying privacy settings and monitoring their children’s behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, there has also been a pushback to this alarmist perspective by scholars who suggest that adolescents’ online privacy should be addressed based on empirical research rather than paternal instinct [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) Empirical evidence from social networks other than TikTok (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Facebook) suggests that adolescents are aware of their social privacy and actively manage their privacy on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As described by boyd [9], adolescents want to avoid surveillance from parents, teachers, friends and other meaningful persons in their lives (that is what “online privacy” means to them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents’ social media use seems to generally prompt increased disclosure of personal information [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, frequent sharing of content does not imply that adolescents share indiscriminately, nor that the content is intended for a wider audience [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Indeed, adolescents are con- cerned about their privacy and capable of protecting it [1, 8, 17, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Contrary to conventional wisdom, young people are, in fact, more likely to protect their privacy on social media than older people [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Madden et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' al found several strategies adolescents use on social media to manage their identity and protect sensitive information [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' These strategies include deleting friends, faking names, delet- ing content, withholding/faking information, and changing privacy settings [20, 35, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They also employed different “zones of privacy” by using different channels for disclosing personal information to maintain intimacy with friends while protecting their privacy from their parents and strangers [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Privacy management can also mean modifying social media content to shield it from audiences [57, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This practice is referred to as “social steganography” or en- coding a message for a defined audience [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents’ privacy management is influenced by various factors such as their social environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', friends, parents), prior (negative) experiences as well as the saliency of privacy settings [53, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Despite the existing evidence on adolescents’ social media use on other social networks, researchers argue that existing findings might not be directly applicable to TikTok [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Compared to other networks such as Facebook or Instagram, TikTok mainly thrives on content exploration and (re)-creation [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The focus is not on the interaction between users and their social network but the interaction with users’ videos proposed by an algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The main feature, the “For You” page, presents an endless stream of personalized, publicly available videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Seeing them will motivate users to react and create similar content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', through features such as “duet” or “stitch”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok might therefore pose a particular threat to adolescents’ privacy because a space previously conceptualized as private and safe can easily become a space of public visibility, surveillance, and judgment (such as in the case of a teenager being seen to perform a dance routine in their bedroom) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Only a few studies have investigated adolescents’ privacy man- agement on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' There is some evidence that privacy manage- ment on TikTok is considered as crucial by adolescents [16] and becomes more stringent at higher perceived risks [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, it is unclear how and why adolescents manage their privacy on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3 COM-B Model As we were interested in the components that shape privacy be- havior, we chose the COM-B model, which has been used in ex- ploratory studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', [32]) and a series of contexts to change behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', [3]), as the conceptual framework for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Many behavioral theories have been developed, often with overlap- ping but differently named constructs [60] and limited guidance on Capability Motivation Opportunity Behavior Reflective Automatic Psychological Physical Social Physical Environment Individual Figure 1: The COM-B model [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The three components capa- bility (C), opportunity (O) and motivation (M) must be present for a behavior (B) to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They interact over time and form a dynamic system with positive and negative feedback loops [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' choosing an appropriate theory for a particular, real-world context [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As a consequence, theories are often under-used to under- stand real-world contexts and to design real-world solutions, which makes replication, implementation, evaluation, and improvements difficult [25, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Researchers have argued that a comprehensive meta-model or “supra-theory” model of behavior – like the COM-B model – is needed that is applicable across contexts [25, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As a meta-model of behavior, the COM-B model does not come with a pre-determined set of context-specific predictions that are common for many behavioral theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' COM-B is based on several exist- ing social cognition models and has a broader understanding of behavior, having "also [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='] automatic processing at its heart [like emotions and habits], broadening the understanding of behaviour beyond the more reflective, systematic cognitive processes that have been the focus of much behavioural research [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='] (for example, social cognition models such as the Theory of Planned Behaviour)" [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Its comprehensive nature and flexibility made it a good fit for the exploratory nature of our study that was not constrained by the conceptual boundaries of a single theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Furthermore, the model comes with hands-on actionable advice on appropriate interventions in a given context in form of a holistic behavior change framework (“Behavior Change Wheel”) (see [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As illustrated in Figure 1, the COM-B model is based on three components – capability (C), opportunity (O), and motivation (M) – that shape a person’s behavior (B) [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Firstly, capability is a subject’s psychological ability (including necessary comprehension, knowledge, and skills) as well as the physical ability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', control of the body) to engage in a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Secondly, motivation can be defined as the subject’s mental processes that energize and di- rect behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It includes the reflective motivation that involves conscious processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', goals, plans, and evaluations) as well as automatic processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', habitual, instinctive, drive-related, and affective processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Finally, opportunity is defined as an attribute of the environmental system (unlike capability and motivation) 3 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' that enables or facilitates a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Opportunities can be physical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', technical features of an app, material, financial, and time) and social (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', norms and culture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In this study, we analyzed the par- ticipants’ capabilities, opportunities, and motivation to engage in privacy behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Firstly, capability is a subject’s psychological ability (including necessary comprehension, knowledge, and skills) as well as the physical ability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', control of the body) to engage in a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Secondly, motivation can be defined as the subject’s mental pro- cesses that energize and direct behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It includes the reflective motivation that involves conscious processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', goals, plans, and evaluations) as well as automatic processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', habitual, instinctive, drive-related, and affective processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Finally, opportunity is de- fined as an attribute of the environmental system (unlike capability and motivation) that enables or facilitates a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Opportunities can be physical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', technical features of an app, material, financial, and time) and social (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', norms and culture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In our exploratory study, we did not focus on identifying inter- actions between COM-B components that explain a specific target behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Rather, our scope was first to learn about the full range of behaviors and explanatory factors associated with adolescents’ privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 3 METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Research Ethics This paper is based on semi-structured, one-to-one interviews with adolescents in the Canton of Zurich, Switzerland, conducted in November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In total, we visited two secondary schools (one in the city of Zurich and one in the greater Zurich area) and three youth centers (all in the city of Zurich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' All interviews were audio- recorded and transcribed verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ethical approval was obtained from our university’s institutional review board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Study participants provided written informed consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For subjects below the age of 16, additional consent was sought from the parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The in- terviews were voluntary and conducted at the institutions from which the subjects had been recruited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Digital, personalized shop- ping vouchers with a value of CHF 20 (~USD 19) were offered to study participants as compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The amount and type of the vouchers was determined beforehand together with the adolescents’ supervisors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', teachers, social workers) in order to not create an inappropriate but still sufficient incentive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' After the interviews, in agreement with the participants, WhatsApp was used to deliver the individualized vouchers to the participants and to allow them to review their personal interview transcripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Several steps were taken to protect the participants identity with- out compromising the transparency of our research process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To begin with, all personally identifiable information was removed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', references to persons, locations) and participants’ names were replaced with pseudonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Furthermore, the study data was stored in line with our university’s storage policy and only the involved re- searchers had access to the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Finally, the original audio files were deleted from all devices half a year after recording together with other remaining personal data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', phone numbers, WhatsApp chats, digital vouchers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Sample and Procedure Due to the lack of research on this topic, we chose a highly ex- ploratory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To identify information-rich cases and make optimal use of available resources, we drew a purposive sample [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We used social media and search engines to find institutions in the Canton of Zurich (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', secondary schools, youth work, youth associations, museums) with contact with adolescents between 12 and 18 years of age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Afterward, principals of participating insti- tutions recruited interested teachers and social workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They, in turn, contacted interested TikTok users in the required age group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To extend the participant base, we applied snowballing among the interested TikTok users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Based on our primary aim (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', to explore how adolescent TikTok users manage their privacy), we chose to sample based on study participants’ age and gender (equally dis- tributed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We decided to ignore other demographic information such as ethnic identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Following a pragmatic definition of theoretical saturation [55], no new information emerged after approximately 40 interviews, and we ended data collection after the 54th interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We chose to employ semi-structured interviews for our study because it encourages two-way communication and provides the interviewer with the opportunity to learn the reasons behind an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some of the questions were part of the interviewer’s guide (see Appendix), others were addressed at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The inter- view guide was developed based on a previous study that applied the COM-B model in a qualitative setting [14] and adopted to the context of the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' After asking for demographic informa- tion, we first explored general TikTok usage and motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The other questions followed the COM-B structure and were related to privacy-related behaviors as well as the explanatory components related to the target behavior “video creation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We finished the interviews with questions about commercial privacy aspects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', targeted advertising and user tracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' After the interview pro- cess was completed, study participants received a copy of their interview transcript via WhatsApp and were invited to add infor- mation or make amendments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Minimal revisions were made by one participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To analyze the content of the interviews, we used a two-step pro- cedure that first divided each statement into one of the four COM-B components (behavior, capability, opportunity, motivation) before further subdividing them into privacy-specific content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For phase one, we used a directed content analysis approach [38] to analyze the statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To counter the subjectivity inherent to qualitative data analysis, three researchers read and coded all statements into the four COM-B domains (behavior, capability, opportunity, motiva- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' On the grounds of economy in both cost and effort, we decided against using "intercoder reliability" (ICR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As full replication of results was deemed unnecessary due to the exploratory and quali- tative nature of data collection and analysis, we instead followed guidelines suggesting the use of "multiple coding" which allows independent researchers to cross check their coding strategies and interpretation of data [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The authors engaged in researcher tri- angulation [21] by discussing the emerging codes during the open coding process of the first three interviews and developed cod- ing guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Disagreements were discussed and resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Using the MAXQDA 2020 software, all responses were coded consistent 4 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) with six COM-B labels1 (behavior, psychological capability, auto- matic/reflective motivation, social/physical opportunity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To ensure continued adherence to the agreed coding guidelines, the three researchers regularly communicated to ensure coding consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In phase two, all statements – previously labeled as one of the COM-B components – were further analyzed for their privacy- specific content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Therefore, an inductive thematic analysis [11] to identify themes within similarly coded statements was con- ducted (see Appendix for coding scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' One researcher identified themes across identically coded statements and discussed them with the other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A theme reflects a collection of similar responses from at least two different study participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For exam- ple, responses that were coded under the COM-B label “reflective motivation” such as “I would be afraid of stupid remarks.”, “I have no desire to be bullied.”, and “I can do without being ridiculed in my class’s WhatsApp group.” were allocated to the privacy-specific theme “negative reaction avoidance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This step resulted in a list of themes within each of the six COM-B labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ultimately, the researchers reviewed and discussed the emerging themes, merged similar themes, and re-labeled others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' By playing the “devil’s ad- vocate” – a common way to scrutinize identified themes [2] – we sought to exploit the full potential of multiple coding to furnish alternative interpretations of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The anonymized, coded interview transcripts are publicly available at osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='io/z8d3w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4 RESULTS A total of 54 adolescents aged between 12-18 years (15 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='82 years) were interviewed, of which half (27) were female (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Interviews ranged from 5 to 21 minutes in length, with a mean of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6 min per interview (SD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Most users attended secondary school, and 80% had used the app for more than one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Half of the study participants admitted using TikTok between one and three hours per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Table 1: Characteristics of one-to-one interview participants (n = 54) Variables % (n) Gender (% of females) 50% (27) Age 15 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='82 Educational level Primary level 2% (1) Lower secondary level 54% (29) Upper secondary level 44% (24) User since One year or less 20% (11) Between one and two years 33% (18) More than two years 46% (25) Current app usage Daily >= 3h 17% (9) Daily >= 1 and <3h 50% (27) Daily < 1h 28% (15) Less than daily 6% (3) 1We did not need to code “physical capability” as the participants did not have physical impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Building on the conceptional framework of the COM-B model, we identified 13 themes from the data analysis that described how and why adolescents protect their privacy on TikTok (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' These are described in more detail in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' No weighting was associated with the themes in terms of their overall contribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Behavior 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Proactive privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The participants in our study mentioned various ways to control the content of their TikToks2 and their audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Publishing content to audiences was described as reflec- tive and non-automatic (as opposed to a habitual, non-reflective publication of TikToks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This behavior is also referred to as the “ap- proach” privacy strategy [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, regarding the content, study participants described what they consider to be too sensitive for publication on TikTok and would not publish (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', TikToks that reveal too much about them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Lima (F, 14) creates public videos and has 50 different accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' She has clear privacy boundaries regarding the video content: “I would not post TikToks where you can see a lot of myself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I wouldn’t post videos in which I’m drunk.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another form of restriction is to define who can see which type of content on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This includes TikTok users making drafts only visible to themselves or blocking selected users from watching videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Bärbel (F, 13) actively tries to keep her parents from seeing her videos: “To prevent my parents from seeing my videos, I can simply block them.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We identified two subthemes within the proactive privacy theme: private creators (19 persons, 35% of the sample) and public creators (11 persons, 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Private creators create videos only for themselves or close friends but do not publish them for a broad audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A few users described the practice of posting videos that are just visible to themselves, only to be able to then repost them on “more private” social media such as Snapchat or WhatsApp for a selected group of people: “I don’t post my videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I download them, save them under photos, then send them on WhatsApp, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I only use TikTok for editing.” (Amy, F, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Public creators regularly create videos for their followers or the general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' An extreme case is Joy (F, 13), who has used TikTok since she was nine years old (when the app was still musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' She maintains 50 thematic user accounts with different age settings and distinct followings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', some accounts for gaming-related videos and others for YouTube reposts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In addition to managing multiple accounts, public creator Lima (F, 14) also uses the live feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It is available to users with at least 1,000 followers and allows them to create personal live streams and interact with users in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Lima had to set her age to 16 years to enable the live feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some study participants reported that they do not publish videos on TikTok at all to protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In the literature, this is referred to as the avoidance privacy strategy [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Peter (M, 14), one of 24 study participants (44%) we classified as a pure consumer, stated: “I’ve never created a TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I don’t even know how to do it.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Tim (M, 12) published once but decided to only watch TikToks afterward: “To try it out, I uploaded something 2The term “TikToks” is used synonymously with videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Table 2: Identified themes for adolescents’ video privacy management on TikTok based on the COM-B model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Frequency is calculated across 54 interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Theme Description Frequency Behavior Proactive privacy Publishing videos with control over the content and the audience 30 Avoidance Publishing no videos on the platform 24 Capability (Psychological) Past privacy incidents Previous negative experiences related to privacy on the platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', lost account, accidental publication) 15 Privacy literacy Knowledge and skills related to privacy management in the app (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', audience understanding and configuration) 53 Opportunity (Social) Negative feedback Negative behavior of others affects privacy management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', observation of cyber-bullying) 16 Linkability experience Observing that online personas can be linked to the personal sphere affects privacy management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', my teacher is on the platform) 39 Restrictive influence Restrictive behavior of others affects privacy management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', restrictive parental mediation) 34 Opportunity (Physical) Platform features Privacy-related features of the platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', audience settings, sharing via other social networks) 46 Device features Privacy-related features of the device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', screen time limits, deleting videos on the smartphone) 17 Motivation (Automatic) Negative emotion avoidance Avoidance of negative emotions expected as a result of publication (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', shame, fear) 15 Motivation (Reflective) Negative reaction avoidance Goal to avoid expected negative consequences of publication 10 Privacy identity Privacy as a general value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', also on other platforms) 5 Publicity avoidance Goal to avoid expected publicity of publication 29 once, but nothing from me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I thought that was funny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' But I prefer to watch videos.” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Capability (Psychological) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Past privacy incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This theme refers to a specific form of privacy-related knowledge (cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [60]) gained after experiencing potential or actual privacy incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Potential privacy incidents are perceived as minor threats but may lead to increased privacy awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' “I posted my very first video by accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It was only seen by three people,” reported Yasmina (F, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Lima (F, 14), a public creator, remembered: “I was half asleep and accidentally posted a TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The next morning, I saw that someone had commented on the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' But I thought it was funny and not bad at all.” When TikTok updated its app and increased the size of the “publish” button to lower the threshold for publication, Lima decided to block app updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Users have also realized that some of TikTok’s privacy features can be easily bypassed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Their awareness of the platform’s weak- nesses has contributed to a greater privacy awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' An example is a feature that allows blocking certain users from viewing videos, which can be easily bypassed: “If I block people but they still want to see my TikToks, they immediately make an extra fake account and continue seeing them.” Roswitha (F, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, she found a way to manage her privacy: “Since these users have too few fol- lowers, I simply block them again or ignore them depending on the video.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A more serious subtheme are actual privacy incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Bärbel (F, 13) had to realize that she was not anonymizing herself sufficiently: “I wore a mask on my face in the video, anonymously, so to speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' But the people who deal with me every day recognized me by my outfit, my room, and my hairstyle and posted the video in the class WhatsApp chat.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Anna (F, 14) reported losing her account and not being able to reclaim it through TikTok’s customer support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' At the same time, other users were still able to watch her videos: “I made videos of myself when I was 9 and then lost the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Now the videos are still public, but I can no longer access them.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Privacy literacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Privacy literacy can be defined as a com- bination of factual or declarative (’knowing that’) and procedural (’knowing how’) knowledge about online privacy [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Concern- ing the publication of videos on the platform, adolescents need to have the knowledge and skills to assess and manage audiences and content as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Respondents mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' that the algorithm might present a video on TikTok’s center stage: “It depends on how pop- ular a video is and only then does it appear on the For You Page.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Bärbel (F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 13)) or that public videos can also be watched without 6 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) having a TikTok account: “From Google or Safari you can type in TikTok and view the videos.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Aron (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They also described how to find out which of their peers used TikTok: “When you post a video, it spreads immediately and then you know who has TikTok and who does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Because so many people have TikTok now, it has become weird for me to post TikToks.” (Elsa (F, 14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Respondents also described their audience and content management skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The private creator Bea (F,14) only publishes for a strictly curated list of followers and therefore has established an approval process that allows her to maintain the desired level of privacy: “I get to know new classmates first and only then give them my TikTok account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Afterward, they tell me they sent a request and I accept them as followers in the app.” (Bea (F, 14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Furthermore, the adolescents interviewed were also able to assess different levels of sensitivity of content in terms of their privacy and select an adequate audience accordingly: “My buddy and I made 10 TikToks in which we share our weekend activities with people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some have 60,000 views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' But we think carefully what to make public.” (Alex (M, 18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The adolescents also talked about various app settings needed to manage the audience, such as the activation of the private account “Switching to the private account takes only two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This is not difficult.” (Alexandra (F, 12)) or knowing the publication status of a video: “A draft is rendered greyish and blurry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' When published, it is bright and jumps right out at you.” (Alexander (M, 15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some adolescents also perform “digital housekeeping” activities by re- moving content related to a specific event or as a habit: “As I became older, I started to delete old videos.” (Ariane (F, 15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3 Opportunity (Social) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Negative feedback refers to expected or observed negative feedback from others (such as harsh comments to videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Study participants reported negative reactions on the platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', from strangers or people from the same school) as an explanation for their privacy protection behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Alexander (M, 15) mentioned a general culture of mutual criticism: “Many of the famous TikTokers sometimes make mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Afterwards, everyone makes fun of them in videos.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Other respondents mentioned nega- tive reactions from their peers that had influenced their behavior: “A friend went viral with a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Then she got yelled at on the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It would annoy me.” (Katja (F, 17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Linkability experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Similar to the perception of negative feedback, the realization of how easily online personas can be linked to the personal sphere can also lead to more restrictive publication behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Study participants perceived the platform as a public space shared by acquaintances and strangers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, by recognizing people from their school on their “For You” page, study participants realized that they, too, could be easily recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As Georg (M, 15) put it: “There are maybe ten or twenty people in the school building who do [public] TikToks regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' You suddenly realize: I know that guy from TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That’s the reason why I don’t publish.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In addition to peers, respondents also described experiences that made them understand that acquainted adults in authority positions would be able to see their TikTok as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Sibylle (F, 15) realized this: “My music teacher was on TikTok singing a song.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Therefore, Sibylle also does not publish so as not to be recognized by everyone on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3 Restrictive influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Restrictive influence refers to others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', close friends or parents) perceived to be restrictive or restrict- ing study participants’ video creation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some interviewees reported that their friends did not publish on TikTok, which in part motivated why they did not publish, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In mentioning his peers, Felix (M, 12) stated: “Most of the people I know don’t upload any- thing of themselves where they show their face.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another example is restrictive mediation by parents or relatives: “My eight-year-old cousin accidentally posted a video with my smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' His uncle saw it on his For You page, so I deleted it.” (Sibylle (F, 15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='4 Opportunity (Physical) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Platform features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Age verification is a key platform feature intended to protect the privacy of young users (not limited to cre- ating videos) and the subject of much public discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In the semi-structured interviews, 29 of the interviewed participants were also asked what age they provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Two-thirds admitted that they had given a false age when they registered (indicating, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', the age of their parents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The main motivation for this behavior was to be able to use TikTok in general (for those below the age of 13) or all its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some study participants, like Martin (M, 14), also had misconceptions about possible age restrictions: “Because otherwise, TikTok won’t let me watch videos.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, study participants also described how they used TikTok’s features for privacy purposes in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This includes using a nick- name instead of their real name, limiting the use of personal infor- mation on their profile page, and not linking their TikTok account with other social media accounts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Instagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While some in- terviewees do not use a name at all: “Why should people know my name?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I have replaced my name and individual letters with an X.” (Ali (M, 12)), others actively involve their parents to make use of the in-app parental controls that restrict their app access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Study participants also reported using various features related to audience configuration, such as creating personal drafts, activating a private account, deleting videos, or blocking users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Public creators sometimes create multiple “privacy-tailored” user accounts with specific follower groups for content of special sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Where the features offered by the platform are perceived as too limited or in- effective, the adolescents used creative workarounds not originally anticipated by the platform provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, it is not easily possible to download and share drafts of videos that are not yet published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Amy (F, 17), however, described a popular workaround: “I post videos on TikTok, but only for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Afterward, I’m able to download them to share them with my friends on WhatsApp.” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Device features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As part of the greater sociotechnical system, some devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', smartphones) offer features that affect user pri- vacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, study participants make use of the “digital well- being” functionality of their smartphone to limit their screentime: “I used TikTok three hours a day because I didn’t know anything better to do with myself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Now I’m trying to get a handle on this with a screen time limit.” stated Matthias (M, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Sandra (F, 14) was one of the study participants who used smartphone features to share videos more selectively: "You can take a screenshot of drafts with an iPhone and then send them via WhatsApp or Snapchat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As mentioned earlier, Lima (F, 14) noticed that the size of the red “publish” button grew with each new app update compared to the 7 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' grey “save as draft” button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Fearing accidental publication, she by- passed this potentially manipulative design pattern (“dark pattern”) by using an old version of the app, which her operating system allowed her to do: “Therefore, I have blocked the updates for TikTok on my cell phone.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='5 Motivation (Automatic) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Negative emotion avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The interviewees describe vari- ous negative emotions if they appeared in a video on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, they mentioned feelings of discomfort, shame, awkward- ness, and annoyance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Milo (M, 12), who does not publish any videos, said: “I would be embarrassed to be seen in a video.” Elsa (F, 14) reported that her desire to avoid negative emotions had evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While she had posted videos on musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly, she didn’t publish on TikTok anymore: “Posting TikToks has become weird for me.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6 Motivation (Reflective) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Negative reaction avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another reason for not pub- lishing personal content was negative reactions by others to their videos such as being bullied in class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', in the WhatsApp class chat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Alexander (M, 15), who does not publish any videos, com- mented: “You make a mistake, people from school see it, it gets sent on, and you get bullied.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Avoidance can also relate to the negative long-term consequences of sharing personal content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As they get older, adolescents who are getting ready to join the job market real- ize that their activity on TikTok could harm their career prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' “The Internet never forgets and if I eventually look for an appren- ticeship, it may be that my future employer sees that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That’s very bad for my reputation.” (Lima (F, 14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Privacy identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' With privacy identity, we refer to a coher- ent set of privacy-related behaviors and personal qualities of an individual in a social setting [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some teenagers consider privacy as a value in itself and part of their identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, for Yara (F, 14), the publication of videos on TikTok is no different from any social network activity: “It’s just not my thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I don’t post in general either, not even on Instagram or anything.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Lena (F, 17) explicitly stated that she considers privacy a significant personal value: “Privacy is important to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I keep everything private that can be kept private.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3 Publicity avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another motivation for restricting the publication of personal videos on the platform is closely related to the linkability experience theme: the desire to not attract public attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Study participants explained that publishing on TikTok means being in the public eye: “It’s a big platform, and I don’t want people around me to see that I make videos.” (Anna (F,14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While in musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly, the public was described as a community of people with similar interests and ages, on TikTok, it is perceived as a heterogenous, superficial place with different people of all ages (including strangers, peers from the same school, teachers, extended family members, and parents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Lina (F, 17) described how the change in the audience had an impact on her behavior: “At musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly, there were also strangers, but more my age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' But TikTok is now worldwide and there are adults everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' I don’t have to post anything there.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Her comment shows that the platform is now perceived as completely public, whereas it used to be a more private community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5 DISCUSSION Our general observation of adolescents’ on TikTok is in line with previous research on other social networks [1, 8, 17, 53]: Contrary to public perception which portrays the publication of TikToks by young people as automatic and unreflective, the adolescents in our sample actively engaged in privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They demon- strated a strong awareness of the need to manage their online identity and social privacy on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, the interview participants were more concerned with protecting their privacy from their immediate social environment than with institutional or commercial privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That is, while they were generally aware that TikTok used algorithms to tailor video content to their partic- ular online behavior, they were more worried about the tangible aspects of the algorithm: that a published video could immediately appear on a classmate’s account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Next, we will discuss the results in more detail following the structure of the COM-B model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While many of our findings are consistent with themes found in previous research on other social media platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Facebook), a few themes and aspects are indeed unique and – best to our knowledge – have not yet been studied by researchers on TikTok or other platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The qualitative nature of our data inform the design of very concrete interventions on TikTok (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Behavior In addition to previous research on other social networks [59], we were able to identify two very different types of proactive pri- vacy behavior: public and private creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While public creators perform privacy management to share videos directly on TikTok, private creators merely use the platform to create and edit videos to share them on other social networks that they see more appro- priate for such content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Snapchat, WhatsApp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It indicates that adolescents have different "imagined audiences" (mental conceptu- alization of the people with whom the user is communicating, [49]) on each social network and curate who sees what by switching between networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A unique finding of our study is that private creators essentially reduce TikTok, which was originally conceived as a social network, to its extensive audio-visual capabilities and share their personal content where social connections already exist and a higher degree of perceived control and intimacy exists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', WhatsApp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It is possible that such a practice might also be found elsewhere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Instagram, YouTube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' At a time when adolescents’ increasingly use multiple social media platforms at once, privacy perceptions of and management between different platforms has to be addressed more comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That is, privacy management can no longer be seen as a single-platform-phenomenon – an obser- vation with important research implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Rather than focusing on isolated social networks with their own privacy standards, re- searchers should expand their analysis to include a cross-network view of privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 8 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Psychological Capabilities Similar to previous studies on other social media platforms [1, 52, 53], we found that adolescents possess knowledge and skills on how to manage their privacy on TikTok (see "privacy literacy" theme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That is, adolescents were not only able to assess the audience of videos but also to actively manage the audience and content of their TikToks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As previously noted [58], privacy management can be very creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This finding also holds true for TikTok: some of our respondents reported using various accounts for different audiences, blocking app updates to avoid receiving less privacy- friendly versions of the app, and making an effort to detect fake users trying to follow them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' An interesting observation that can potentially inform other research on social privacy management in social networks is that adolescents on TikTok do not only use the technical features provided by the social network itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Instead, some are also capable of using physical opportunities provided the device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', blocking app updates, screen time management).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This example illustrates how the existence of these generic physical opportunities provided by the operating system can influence the privacy management capability of young TikTok users to learn about additional ways to protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In line with previous research we found that negative past expe- riences affect future privacy management behaviors [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Incidents can even serve as a learning opportunity [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In our sample, partici- pants experienced near or actual privacy incidents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', accidentally publishing videos, loss of account with personal videos) that led them to adapt their privacy management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', immediately deleting accidentally published videos, paying more attention to a publi- cation in the future).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While our data support the hypothesis that incidents serve as learning opportunities, it must be said that cer- tain very extreme violations of privacy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', persistent bullying or stalking) have not been reported in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It is unclear how such experiences affect privacy behavior in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Nonetheless, our findings inform future research by showing that even minor pri- vacy incidents without severe consequences can lead to improved capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3 Physical Opportunities Adolescents in our sample used various features of TikTok and the operating system to manage their privacy (themes platform features and device features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' At the same time, they were aware of TikTok’s privacy management limitations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', the ineffective- ness of blocking users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some of the measures TikTok has taken to protect the privacy of younger users in response to public criticism may not be very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Out of 29 study participants with whom we discussed the topic, two-thirds used a false age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Many teenagers we interviewed have been publishing on TikTok much before the legally allowed age of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Regardless of the normative standpoint, this calls into question TikTok’s fine-grained, age-based privacy features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Despite legislative measures such as the Children’s Online Privacy Protection Act of 1998, this problem has been described on other social networks in the past [51, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Sometimes also parents help their underage children to access social networks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Rea- sons for using social networks below the specified minimum age are diverse (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', wanting to stay in touch with classmates, want- ing unrestricted access to TikTok’s features) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Consequently, technical measures to protect children such as non-public accounts or content restrictions are failing [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' boyd et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' al [10] called for abandoning ineffective age-based mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Instead, she advo- cates for an honest discussion about children’s use of social media and a rethinking of the industry to better incorporate the needs of children and parents when developing apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another issue on social networking sites is account loss [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This issue was also highlighted by several of our respondents who reported that they were unable to reclaim a video they had posted after losing an account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As a consequence, they were unable to revoke their consent from publishing a childhood experiment that would now remain online forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This is particularly problematic against the background of increasingly better algorithms for rec- ognizing people in images and videos and the resulting linkability risk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Clearview AI [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To exercise the "right to be forgotten" as embodied in the EU GDPR, for example, the ability to reclaim accounts and delete old videos is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It is unclear whether ac- count loss among adolescents is a broader phenomenon or whether other social networks are affected as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='4 Social Opportunities Our findings on TikTok support previous research demonstrating that the social environment of teenagers shapes their privacy be- haviors [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Other social network users as well as the parents are major agents of socialization [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Social norms, which emerge as a response to observed behavior or expected attitudes of friends and parents, influence children’s intention to share personal infor- mation [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' If friends and parents disapprove of such behavior, children tend to share less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A recent study on TikTok described, that restrictive mediation by parents can also lead to more restrictive disclosure behavior in children [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In our study, we identified similar social influences on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Observing strangers being publicly criticized for videos (theme negative feedback) resulted in restrictive publication behavior by the adolescents we interviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In line with previous research [77], the restrictive norms and behavior of relatives, parents, and friends were also found to have the potential to affect behavior on TikTok (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', not publishing or blocking parents from videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' What makes TikTok stand out from other social networks, is its specific content algorithm based on a granular observation of user preferences [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The results of our study indicate that prevalent TikTok usage among peers in combination with the platform’s spe- cific algorithm that immediately displays the published content to cohorts with similar attributes – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', peers – may increase the social influence of others on adolescents’ privacy behavior (“linkability experience”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Unlike posting a video under a nickname on YouTube that may never be discovered by peers, adolescents were aware that posting on TikTok was potentially more privacy-invasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They recognized that their videos could become visible to their personal environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', in the schoolyard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This experience led to re- stricted publication behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='5 Automatic and Reflective Motivations Adolescents’ motivations for protecting their privacy on TikTok were based on either wanting to avoid publicity, to avoid nega- tive reactions/emotions, or to actively achieve privacy (themes 9 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' negative reaction avoidance, publicity avoidance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The adolescents interviewed reported wanting to evade the public eye and feared negative feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', public criticism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' These are themes previ- ously described on other social networks [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To avoid a negative emotional outcome (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', shame), they refrain from having a too public profile (theme negative emotion avoidance) (see [13] for a similar finding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For some adolescents, privacy was a personal matter beyond TikTok (theme privacy identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That is, these teenagers were in- trinsically motivated to keep their information private - a finding that stands in contrast with previous research on other social net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Research suggests that, on average, adolescents have fewer privacy concerns than young adults [4, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, our findings indicate that these concerns can vary greatly across adolescents, and some may place great value on their privacy on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Even though the theme was mentioned by only a few participants, it underscores that adolescents are not a homogeneous group when it comes to motives for managing privacy on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For some participants’ being private is a personal value and their goal is to achieve a coherent privacy behavior on TikTok and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='6 Methodological Consideration For our study, the COM-B model helped to holistically understand adolescents’ privacy management on TikTok related to the creation of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It has a solid theoretical foundation and – according to its authors – can be applied across various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, much of the research to date has applied the COM-B model to health-related behaviors such as smoking cessation and lowering cardiovascular disease risk [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our study, which showed that the COM-B model is also a suitable analytical framework for studying privacy behavior, provides yet another use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' By demonstrating its relevance to the privacy management of adolescents, we strengthen the model’s extrinsic validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='7 Possible Approaches for Privacy Interventions Several of the themes we identified can be used as starting points for the development of privacy interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The COM-B model is part of a theory-driven intervention development framework called behavior change wheel (BCW), a synthesis of behavior change frameworks [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In the logic of the BCW, interventions are di- rected at desired “target behaviors” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', enabling privacy settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Building on the interview findings and our observations, Figure 2 shows different parties and ideas for potential target behaviors af- fecting adolescents’ video privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It focuses on which behaviors to address and does not answer the question of how to design interventions that address these behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', adequate behavior change techniques [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Any intervention schemes to improve the privacy of adoles- cent TikTok users should focus on the behavior of the adolescents themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The interviews provide concrete suggestions for be- haviors that adolescents already report that improve their privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This includes encouraging young users to remove in- appropriate videos from the platform and the use of alternative social media apps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', WhatsApp) to share content (theme: proac- tive privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some of our participants reported regular checks if a video with the status “published” should be set to private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' They also removed their old TikToks from the app and their smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our private creators did seldom publish on TikTok but used alternative apps such as Snapchat or WhatsApp with a perceived higher level of privacy and the ability to automatically delete shared TikToks after being watched by their friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another possible target behav- ior derived from our observations is “backing up user credentials” (theme: privacy literacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some adolescents in our sample who had already created accounts in musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ly could not delete published videos because they had forgotten their credentials, and were not able to prove their identity to the TikTok support to retrieve their account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' An intervention could mitigate account loss, especially in cases where children have multiple accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Finally, teenagers should be made aware of the privacy settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', the private ac- count) and the potential risks of not correctly setting these (theme: reflective motivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, in our interviews, participants accidentally published a TikTok upon their first usage of the app because they were not aware others would immediately see it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The platform must also play an important role in safeguarding the privacy of children and adolescents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Improving features directly related to privacy such as improved age verification, more effec- tive blocking of users, and facilitating access to lost user accounts are promising approaches (theme: platform features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As described earlier, many adolescents in our sample did not use their real age due to various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, they were often unaware that the private account would have been activated by default if they had provided their real age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As a result of providing false infor- mation, the privacy settings were much more lenient and TikTok videos would not only be published to followers but to everybody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Following boyd et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' al’s [10] philosophy, one possibility would be to abandon TikTok’s age-based mechanism and incorporate the needs of children and parents when developing the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For TikTok, this could mean taking a certain level of responsibility for its con- tent and giving kids and parents ways to control what videos are shown (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', via a content configuration or a separate app similar to YouTube Kids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Even if the app adhered to the age-based privacy concept, describing the consequences of providing real age (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', better privacy protection) might encourage some youth to provide their real age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Another approach has been lately launched by the twin app Douyin [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Douyin introduced an age verification that is not based on self-declaration only but requires – unlike the in- ternational counterpart TikTok - user authentication and imposes restrictions on the permitted daily use for users under 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some participants also criticized that they could not effectively block users who they wanted to prevent from seeing their videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The problem persists because blocked users can immediately "respawn" under a different username.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok could prevent this issue with a feature that block all accounts of the same user (similar to Insta- gram [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Some study participants also reported feeling “nudged” by the user interface design towards publishing TikTok video for a broad audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Others described publishing personal TikToks acci- dentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While nudging teenagers towards better privacy behavior is also controversial [78], presenting them with simple alternatives (such as publishing a TikTok vs saving a local draft) could provide a welcome middle ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Furthermore, TikTok might also do more to educate its users on how to protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This suggestion 10 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) TikTok Users Family & Friends Schools & Youth Work Policy- Makers & Privacy Advocates Other Platform Users OS Vendors & Other Apps ByteDance (TikTok Creator) Enable privacy settings Share/remove personal content consciously Backup account credentials Share TikToks using other apps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', Whatsapp) Inform each other about privacy possiblities Do not nudge users towards publication Educate children about long- term privacy risks Improve privacy features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', reclaiming ‘lost’ accounts, age verification) Use TikTok yourself to understand privacy issues Educate students early about longterm privacy risks Support childrens’ privacy efforts Provide privacy tutorials Create & enforce privacy laws (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', transparency about PII usage, age verification) Housekeeping functionality for TikTok Enforce app privacy in OS Explain business model of TikTok Use TikTok yourself to understand privacy issues Explain business model of TikTok Respect the privacy of others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', norms) Figure 2: Different parties and their potential target behaviors relevant for adolescents’ video privacy management on TikTok is based on our observation that capabilities varied between adoles- cents and TikTok users had begun to create such privacy tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The latter indicates a demand for more support (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', via privacy tutorials provided by TikTok).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Family, friends, schools, and youth workers can also positively in- fluence the privacy management of adolescents (social opportunity themes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In addition to supporting adolescents’ privacy efforts, their social network could use TikTok themselves to better understand specific privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In our sample, an uncle of an eight-year-old boy used TikTok himself and warned him about the possibility on TikTok of publishing a video by accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The social environment can also advise about long-term privacy risks to the children and adolescents of which they might not yet be aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Among a group of adolescents of the same class, we repeatedly heard the narrative of a classmate being recognized on TikTok despite her wearing a mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Due to this “risk narrative” the whole class was aware of the potential risks of insufficient anonymization on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A collection of such tales could be used by teachers in the classroom to illustrate the privacy risk associated with the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As users do not only interact with each other when they share videos but also with the platform and its owner company, teenagers should also be made aware of commercial privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our data confirmed that adolescents’ primary privacy focus was indeed so- cial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To this end, adolescents would need to understand TikTok’s business model, which heavily relies on their personal data, and the organization behind TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Policymakers and privacy advocates are also relevant actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Not only do they seek to create privacy laws to protect users but also to enforce these laws through, for example, insisting on effective age verification (theme: platform features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ideally, these actions are guided by evidence in collaboration with researchers, adolescent users, and parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' For example, our findings indicate that ado- lescents did not know that TikTok had taken additional measures to protect them in 2021 [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' While privacy legislation demands transparency for data subjects – especially for children – this ex- ample shows that there is room for improvement in terms of the implementation of laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' It should also be mentioned that other TikTok users can influence an adolescent’s privacy behavior (social opportunity themes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Older and more experienced teenagers may have capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', based on their negative experiences) that can benefit younger and less experienced users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' One of our participants reported having learned about privacy settings from a video on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Indeed, some more experienced users have already begun to acts as mentors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' This includes the user @seansvv with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 million followers, who stated in his biography “I Read ToS [Terms of Service] So That You Don’t Have To” and regularly posts TikTok videos related to privacy topics [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Finally, our interviews showed that OS vendors and the vendors of other apps contribute to teenagers’ privacy on TikTok (theme: device features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' OS vendors have implemented more and more privacy control mechanisms for their end-users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', granular rights management, location sharing notifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' These methods all work on low-level personal data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', IP address, location, and email address).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' However, videos shared by adolescents on TikTok that possibly contain more sensitive personal data with higher risks involved are not yet covered by these mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' At times when a user publishes a video accidentally, the OS could warn them in the same way that they are warned when sharing their location with the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In our sample, participants reported also manually cleaning 11 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' up their TikToks in the app and on their phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' OS vendors could provide housekeeping functionalities that would simplify removing personal content across different social networks and on the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='8 Limitations and Future Research As with most qualitative research, our sample is small and was not drawn randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Therefore, we cannot claim that the results are representative of all young people in the region under consideration, and certainly not of Switzerland as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Further validation with different samples is needed to strengthen the findings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', including subjects’ socioeconomic status).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Choosing interviews as our data collection methodology was use- ful to learn more about the perspectives of adolescents in Switzer- land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Nevertheless, we are aware of the limitations associated with this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Primarily, we relied on self-reporting rather than be- havioral observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Self-reports can be biased due to various influences, such as subjects’ desire to portray themselves in a posi- tive light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Future studies might want to gather data from a wider range of sources, such as direct observations of privacy manage- ment behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', through TikTok data donations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Based on our findings, future research could develop and system- atically test privacy interventions based on the BCW methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A necessary first step would be to identify appropriate target be- haviors with the greatest potential to improve privacy management among adolescents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our research could be a starting point for select a “promising” target behavior reported by the adolescents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', ac- tivating the private account) to address in a target population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', pupils of a local school).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To identify a baseline for each of the poten- tial behaviors and to select a target behavior among them, further research would be necessary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', in form of a survey among pupils).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Furthermore, additional research is required to select appropriate behavior change techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', increasing awareness for privacy settings) and evaluate their effectiveness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', with an experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Importantly, such research could also control for factors such as socioeconomic status might also be relevant to explain privacy- related behaviors on TikTok [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Given that teenagers may have very heterogeneous privacy management capabilities, motivations, and opportunities, depending on their age and experience regarding the platform, interventions need to be tailored to the specific target group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Large-scale intervention studies using the BCW can help to identify effective and evidence-based policies to improve privacy management among young people on social media platforms like TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Our interviews focused on social aspects of adolescents’ privacy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' That is, our interviewees were more concerned with protecting their privacy from their social environment than from the corporations dealing with their data for commercial purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' see [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Yet, TikTok videos are not only shared with other users but also with ByteDance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Even the users we identified as pure consumers who only view but not create content may have privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' As the video and ad algorithms are known for their high level of customization, they make the platform heavily reliant on personal data including detailed user behavior [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Both users’ active and passive behavior on the app has consequences: The TikTok pixel allows companies to engage in detailed web tracking of TikTok users on websites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=', a user who sees the ad on TikTok might buy the product in the online shop) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Further research could investigate if adolescent users are aware of these commercial privacy aspects and how they manage them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' ACKNOWLEDGMENTS The research reported in this article was funded by the Digital Future Fund (DFF), which is part of the Digitalization Initiative of the Zurich Higher Education Institutions (DIZH), Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We would like to thank all adolescents, teachers, and social workers we contacted in conducting our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' We also thank Frank Wieber, Katja Kurz and Manuel Günther for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' REFERENCES [1] Claire Balleys and Sami Coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Being Publicly Intimate: Teenagers Managing Online Privacy.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' JAMES: Jugend, Aktivitäten, Medien–Erhebung Schweiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Working Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Zurich University of Applied Sciences, Winterthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Big Data & Society 9, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1177/20539517211065368 [14] Oscar Castro, Ineke Vergeer, Jason Bennie, Jonathan Cagas, and Stuart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Biddle.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Pitts, Nick Steen, Ruth Thomas, Anne Walker, and Marie Johnston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Explaining Clinical Behaviors Using Multiple Theoretical Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Implementation Science 7, 1 (2012), 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': 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+page_content='com/article/emilybakerwhite/tiktok- tapes-us-user-data-china-bytedance-access [27] Ilker Etikan, Sulaiman Abubakar Musa, Rukayya Sunusi Alkassim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Comparison of Convenience Sampling and Purposive Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' American journal of theoretical and applied statistics 5, 1 (2016), 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [28] European Court of Human Rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Factsheet – Right to the Protection of One’s Image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='echr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='coe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='int/documents/ fs{_}own{_}image{_}eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='pdf [29] European Data Protection Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Italian DPA Imposes Limita- tion on Processing on TikTok after the Death of a Girl from Palermo | European Data Protection Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://edpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='eu/news/national-news/2021/italian-dpa-imposes- limitation-processing-tiktok-after-death-girl-palermo{_}pl [30] Coco Feng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Chinese Version of TikTok Gets 40-Minute Time Limit for Kids under 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='scmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/tech/policy/article/ 3149397/chinese-version-tiktok-limits-kids-under-14-40-minutes-day-adding- fight [31] Yang Feng and Wenjing Xie.' metadata={'source': 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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='009 [32] Caragh Flannery, Sheena McHugh, Ann Ebere Anaba, E Clifford, Mairead O’Riordan, Louise C Kenny, Fionnuala M McAuliffe, Patricia M Kearney, and M Byrne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Enablers and barriers to physical activity in overweight and obese pregnant women: an analysis informed by the theoretical domains framework and COM-B model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' BMC pregnancy and childbirth 18, 1 (2018), 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [33] FTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Video Social Networking App Musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='Ly Agrees to Settle FTC Allegations That It Violated Children’s Privacy Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11- 02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ftc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='gov/news-events/press-releases/2019/02/video-social- networking-app-musically-agrees-settle-ftc [34] Eszter Hargittai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Digital Na(t)Ives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Variation in Internet Skills and Uses among Members of the “Net Generation”*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Sociological Inquiry 80, 1 (2010), 92–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1475-682X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='00317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='x [35] Wannes Heirman, Michel Walrave, Anne Vermeulen, Koen Ponnet, Heidi Van- debosch, Joris Van Ouytsel, and Ellen Van Gool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' An Open Book on Facebook?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Examining the Interdependence of Adolescents’ Privacy Regula- tion Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Behaviour & Information Technology 35, 9 (2016), 706–719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1080/0144929X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1181210 [36] Kashmir Hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Secretive Company That Might End Privacy as We Know It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/2020/01/18/technology/clearview-privacy-facial- recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='html [37] Lisa Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Parent’s Guide to TikTok - Everything You Need to Know to Keep Your Child Safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='dailyrecord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' uk/lifestyle/family-kids/parents-guide-tiktok-everything-you-21632471 [38] Hsiu-Fang Hsieh and Sarah E Shannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Three Approaches to Qualitative Content Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Qualitative health research 15, 9 (2005), 1277–1288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [39] Instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' What Happens When I Block Someone on Instagram?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/447613741984126?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='locale=en{_}US& cms{_}id=447613741984126&published{_}only=true [40] Hyunjin Kang, Wonsun Shin, and Junru Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Teens’ Privacy Management on Video-Sharing Social Media: The Roles of Perceived Privacy Risk and Parental Mediation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Internet Research 32, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [41] Jacob Kastrenakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Now Lets Parents Make Their Teens’ Accounts More Private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='theverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/2020/11/17/21570244/tiktok-parental-controls- family-pairing-private-accounts-search-limits [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Bondy Valdovinos Kaye, Xu Chen, and Jing Zeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Co-Evolution of Two Chinese Mobile Short Video Apps: Parallel Platformization of Douyin and TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Mobile Media & Communication 9, 2 (2021), 229–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 1177/2050157920952120 [43] Melanie Kennedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' ‘If the Rise of the TikTok Dance and e-Girl Aesthetic Has Taught Us Anything, It’s That Teenage Girls Rule the Internet Right Now’: TikTok Celebrity, Girls and the Coronavirus Crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' European Journal of Cultural Studies 23, 6 (2020), 1069–1076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1177/1367549420945341 [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Laeeq Khan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Social Media Engagement: What Motivates User Participa- tion and Consumption on YouTube?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Computers in Human Behavior 66 (2017), 236–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='024 [45] Daniel Klug, Yiluo Qin, Morgan Evans, and Geoff Kaufman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Trick and Please.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A Mixed-Method Study On User Assumptions About the TikTok Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In 13th ACM Web Science Conference 2021 (WebSci ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 84–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1145/3447535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='3462512 [46] Felix Krause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' iOS Privacy: Announcing InAppBrowser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='Com - See What JavaScript Commands Get Injected through an in-App Browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://krausefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/blog/announcing-inappbrowsercom-see- what-javascript-commands-get-executed-in-an-in-app-browser [47] Dami Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Popular Musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='Ly App Has Been Rebranded as TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Re- trieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='theverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/2018/8/2/17644260/musically- rebrand-tiktok-bytedance-douyin [48] Dami Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Stops Young Users from Uploading Videos after FTC Settlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='theverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/2019/2/27/ 18243510/tiktok-age-young-user-videos-ftc-settlement-13-childrens-privacy- law [49] Eden Litt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Knock, Knock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Who’s There?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Imagined Audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Journal of Broadcasting & Electronic Media 56, 3 (2012), 330–345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1080/ 08838151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='705195 [50] Sonia Livingstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Taking risky opportunities in youthful content creation: teenagers’ use of social networking sites for intimacy, privacy and self-expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' New media & society 10, 3 (2008), 393–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [51] Sonia Livingstone, Leslie Haddon, Anke Görzig, and Kjartan Ólafsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Risks and Safety on the Internet: The Perspective of European Children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' LSE, London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [52] Sonia Livingstone and Ellen Helsper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Balancing Opportunities and Risks in Teenagers’ Use of the Internet: The Role of Online Skills and Internet Self- Efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' New Media & Society 12, 2 (2010), 309–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1177/ 1461444809342697 [53] Sonia Livingstone, Mariya Stoilova, and Rishita Nandagiri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Children’s Data and Privacy Online: Growing up in a Digital Age: An Evidence Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' London School of Economics and Political Science, Department of Media and Communications, London, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='lse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='uk/my-privacy-uk [54] Sonia Livingstone, Kjartan Ólafsson, and Elisabeth Staksrud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Social Net- working, Age and Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' LSE, London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [55] Jacqueline Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A Pragmatic Definition of the Concept of Theoretical Saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Sociological Focus 52, 2 (2019), 131–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1080/ 00380237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1544514 [56] ByteDance Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Creating Videos | TikTok Help Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/en/using-tiktok/creating-videos [57] Mary Madden, Amanda Lenhart, Sandra Cortesi, Urs Gasser, Maeve Duggan, Aaron Smith, and Meredith Beaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Teens, Social Media, and Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Pew Research Center 21, 1055 (2013), 2–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [58] Alice E Marwick and danah boyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Networked Privacy: How Teenagers Negotiate Context in Social Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' New Media & Society 16, 7 (2014), 1051–1067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1177/1461444814543995 [59] Alice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Marwick, Diego Murgia-Diaz, and John G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Palfrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Youth, Privacy and Reputation (Literature Review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' SSRN Scholarly Paper ID 1588163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Social Science Research Network, Rochester, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/abstract= 1588163 [60] Susan Michie, Lou Atkins, and Robert West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2014-05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Behaviour Change Wheel: A Guide to Designing Interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Silverback Publishing, Sutton, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [61] Susan Michie, Michelle Richardson, Marie Johnston, Charles Abraham, Jill Francis, Wendy Hardeman, Martin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Eccles, James Cane, and Caroline E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Behavior Change Technique Taxonomy (v1) of 93 Hierarchically Clustered Techniques: Building an International Consensus for the Reporting of Behavior Change Interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Annals of Behavioral Medicine 46, 1 (2013), 81–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https: 13 Proceedings on Privacy Enhancing Technologies 2023(2) Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1007/s12160-013-9486-6 [62] Susan Michie, Maartje M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' van Stralen, and Robert West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Behaviour Change Wheel: A New Method for Characterising and Designing Behaviour Change Interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Implementation Science 6, 1 (2011), 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 1186/1748-5908-6-42 [63] Christian Montag, Haibo Yang, and Jon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Elhai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' On the Psychology of TikTok Use: A First Glimpse From Empirical Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Frontiers in Public Health 9 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [64] Caron Mullen and Nicola Fox Hamilton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents’ Response to Parental Facebook Friend Requests: The Comparative Influence of Privacy Management, Parent-Child Relational Quality, Attitude and Peer Influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Computers in Human Behavior 60 (2016), 165–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='026 [65] Brian O’Neill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Who Cares?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Practical Ethics and the Problem of Underage Users on Social Networking Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ethics and Information Technology 15, 4 (2013), 253–262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1007/s10676-013-9331-4 [66] Shailendra Rathore, Pradip Kumar Sharma, Vincenzo Loia, Young-Sik Jeong, and Jong Hyuk Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Social Network Security: Issues, Challenges, Threats, and Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Information Sciences 421 (2017), 43–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='063 [67] Kate Raynes-Goldie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Aliases, Creeping, and Wall Cleaning: Understanding Privacy in the Age of Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' First Monday 15, 1 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5210/fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='v15i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2775 [68] Kalhan Rosenblatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Surpasses Google as Most Popu- lar Website of the Year, New Data Suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11- 02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='nbcnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/tech/tech-news/tiktok-surpasses-google- popular-website-year-new-data-suggests-rcna9648 [69] Greg Roumeliotis, Yingzhi Yang, Echo Wang, and Alexandra Alper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Ex- clusive: U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Opens National Security Investigation into TikTok - Sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='reuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/article/us-tiktok-cfius- exclusive-idUSKBN1XB4IL [70] Fergus Ryan, Audrey Fritz, and Daria Impiombato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Privacy Concerns and Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Australian Strategic Policy Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 36–42 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' jstor:resrep26120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='7 [71] SEAN [They/Them] (@seansvv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/@seansvv [72] Wonsun Shin and Hyunjin Kang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents’ Privacy Concerns and Information Disclosure Online: The Role of Parents and the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Computers in Human Behavior 54 (2016), 114–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='062 [73] Statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Annual Installs 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/statistics/1089420/tiktok-annual-first-time-installs/ [74] Statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Users by Age 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/statistics/1095186/tiktok-us-users-age/ [75] Terri Peters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' How TikTok’s New Kids’ Privacy Settings Will Change the Way They Use It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/parents/tiktok- changes-privacy-settings-kids-under-18-t205733 [76] Sabine Trepte, Doris Teutsch, Philipp K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Masur, Carolin Eicher, Mona Fischer, Alisa Hennhöfer, and Fabienne Lind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Do People Know About Privacy and Data Protection Strategies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Towards the “Online Privacy Literacy Scale” (OPLIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In Reforming European Data Protection Law, Serge Gutwirth, Ronald Leenes, and Paul de Hert (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Springer Netherlands, Dordrecht, 333–365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1007/978-94-017-9385-8{_}14 [77] Ellen Van Gool, Joris Van Ouytsel, Koen Ponnet, and Michel Walrave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' To Share or Not to Share?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Adolescents’ Self-Disclosure about Peer Relationships on Facebook: An Application of the Prototype Willingness Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Computers in Human Behavior 44 (2015), 230–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='036 [78] Mariana Veretilnykova and Leyla Dogruel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Nudging Children and Adoles- cents toward Online Privacy: An Ethical Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Journal of Media Ethics 36, 3 (2021), 128–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1080/23736992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1939031 [79] Echo Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Hits 1 Billion Monthly Active Users Globally - Com- pany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='reuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/technology/tiktok-hits-1-billion-monthly- active-users-globally-company-2021-09-27/ [80] Robert West and Susan Michie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' A Brief Introduction to the COM-B Model of Behaviour and the PRIME Theory of Motivation [V1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='qeios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/read/WW04E6 [81] Alan F Westin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Privacy and Freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Athenum, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' [82] Pamela Wisniewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2018-03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' The Privacy Paradox of Adolescent Online Safety: A Matter of Risk Prevention or Risk Resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' IEEE Security Privacy 16, 2 (2018-03), 86–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1109/MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1870874 [83] Pamela J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Wisniewski, Jessica Vitak, and Heidi Hartikainen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Privacy in Adolescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In Modern Socio-Technical Perspectives on Privacy, Bart P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Knij- nenburg, Xinru Page, Pamela Wisniewski, Heather Richter Lipford, Nicholas Proferes, and Jennifer Romano (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Springer International Publishing, Cham, 315–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1007/978-3-030-82786-1{_}14 [84] Queenie Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' TikTok Accused of Secretly Gathering User Data and Send- ing It to China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Retrieved 2022-11-02 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='cnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='com/news/privacy/ tiktok-accused-of-secretly-gathering-user-data-and-sending-it-to-china/ [85] Alyson Leigh Young and Anabel Quan-Haase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Privacy Protection Strategies on Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Information, Communication & Society 16, 4 (2013), 479–500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1080/1369118X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='777757 [86] Brahim Zarouali, Karolien Poels, Koen Ponnet, and Michel Walrave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' “Ev- erything under Control?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=': Privacy Control Salience Influences Both Critical Processing and Perceived Persuasiveness of Targeted Advertising among Ado- lescents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Cyberpsychology: Journal of Psychosocial Research on Cyberspace 12, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='5817/CP2018-1-5 [87] Diana Zulli and David James Zulli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Extending the Internet Meme: Concep- tualizing Technological Mimesis and Imitation Publics on the TikTok Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' New Media & Society 24, 8 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1177/1461444820983603 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='1 Interview Guide (translated from German) (1) What’s your first name?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' How old are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' In what grade are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (2) How often do you use TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' How long have you been using TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' When did you start to use TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (3) Do you remember how old you were when you started using the app?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (4) How many people do you follow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' How many followers do you have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (5) Do you share videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' How many?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' What types of videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Behavior) (a) Why do you/don’t you share videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Motivation) (b) If yes: How do you share videos on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Psychological capability) (6) Who can see your videos when they are shared?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Psycholog- ical capability) (7) How can you influence who can see your videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Psycho- logical capability) (8) Do you restrict who can see your videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Behavior) (a) If yes: Why / When do you restrict your videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Motiva- tion) (b) If no: Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Have you ever considered restricting your videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Motivation) (9) Do your friends or others restrict their/your videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Social Opportunity) (10) Have you ever accidentally posted a video?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' If yes: What did you do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Psychological capability) (11) What do you think about TikTok’s features to share/restrict videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' (Physical opportunity) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content='2 Coding Scheme Table 3 shows the hierarchical coding scheme together with the frequency of each code calculated across the 54 interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 14 Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok Proceedings on Privacy Enhancing Technologies 2023(2) Table 3: Coding Scheme (translated from German).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Frequency is calculated across 54 interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' Code Description Frequency Usage_since Start of TikTok usage 54 Usage_frequency Frequency of TikTok usage 53 App_age Age entered into the app at first use 29 Video_Behavior Avoidance I normally do not create/publish TikToks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 24 Proactive PersonalCreator I regularly create/publish TikToks for myself and close friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 19 PublicCreator I regularly create/publish TikToks for my followers/the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 11 Video_PsyCapability PastPrivacyIncidents Minor I have perceived a potential/minor privacy incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 9 Severe I have perceived a severe privacy incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 8 PrivacyLiteracy AudienceContentLiteracy I’m aware of different audience/content types and have the ability to manage them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 52 TechnicalLiteracy I have the technical knowledge and skills to manage my audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 50 Video_SocOpportunity NegativeFeedback Others show negative reactions to TikToks, that’s why I’m not active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 16 LinkabilityExperience Users can be easily recognized in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 39 RestrictiveInfluence I’m not active because others are also restrictive or enforce my privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 34 Video_PhyOpportunity PlatformFeatures TikTok helps to ensure my privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 46 DeviceFeatures The device helps to ensure my privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 17 Video_AutMotivation NegativeEmotionAvoidance I don’t publish content to avoid negative emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 15 Video_RefMotivation NegativeReactionAvoidance I don’t publish content to avoid negative reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 10 PrivacyIdentity I don’t publish content because privacy is important to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 5 PublicityAvoidance I don’t publish content because I don’t want to be in the public eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} +page_content=' 29 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFJT4oBgHgl3EQfpiwj/content/2301.11600v1.pdf'} diff --git 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DOYLE AND DAVID KRUMM +Abstract. Among all the dynamical modular curves associated to quadratic polynomial +maps, we determine which curves have infinitely many quadratic points. +This yields a +classification statement on preperiodic points for quadratic polynomials over quadratic fields, +extending previous work of Poonen, Faber, and the authors. +1. Introduction +Let K be a field with algebraic closure K, and let f be a rational function in one variable +over K. Corresponding to f there are a morphism of algebraic varieties P1 +K → P1 +K and a +map on point sets P1(K) → P1(K), both of which we also denote by f. A point P ∈ P1(K) +is called periodic for f if there exists a positive integer n such that f n(P) = P, where f n +denotes the n-fold composition of f with itself; in that case, the smallest such n is called +the (exact) period of P. More generally, the point P is preperiodic for f if there exists +m ≥ 0 such that f m(P) is periodic; the smallest such m is then called the preperiod of P, +and we call the period of f m(P) the eventual period of P. Here, f 0 is interpreted as the +identity map, so that periodic points are considered preperiodic. +For any intermediate field K ⊆ L ⊆ K we define a directed graph G(f, L), called the +preperiodic portrait of f over L, whose vertices are the points P ∈ P1(L) that are +preperiodic for f, and whose directed edges are the ordered pairs (P, f(P)) for all vertices +P. In the terminology of graph theory, G(f, L) is a functional graph, i.e., a directed graph in +which every vertex has out-degree 1. Throughout this paper we will use the term portrait +instead of functional graph in order to emphasize our dynamical perspective. +1.1. Portraits for quadratic maps. Assume henceforth that K is a number field. We will +primarily, though not exclusively, be interested in the case where K is a quadratic extension +of Q; we refer to such fields simply as quadratic fields. A type of problem that has received +much attention in the field of arithmetic dynamics is that of classifying the portraits G(f, K) +up to graph isomorphism as f is allowed to vary in an infinite family of rational functions. +An early example of this classification problem is Poonen’s study [43] of the portraits G(f, Q) +as f varies over the family of all quadratic polynomials with rational coefficients. +Theorem 1.1 (Poonen [43]). Assume that there is no quadratic polynomial over Q having +a rational periodic point of period greater than 3. +Then, for every quadratic polynomial +f ∈ Q[z], the portrait G(f, Q) is isomorphic to one of the following twelve graphs (using the +labels from Appendix B): +∅, 2(1), 3(1, 1), 3(2), 4(1, 1), 4(2), 5(1, 1)a, 6(1, 1), 6(2), 6(3), 8(2, 1, 1), 8(3). +Date: January 3, 2023. +2020 Mathematics Subject Classification. Primary 37P05, 37P35; Secondary 37P15, 11G30, 14G05. +Key words and phrases. Arithmetic dynamics, dynatomic curve, preperiodic portrait, uniform bounded- +ness conjecture. +1 +arXiv:2301.00510v1 [math.NT] 2 Jan 2023 + +2 +JOHN R. DOYLE AND DAVID KRUMM +Regarding the assumption in Theorem 1.1, it is known that a quadratic polynomial over +Q cannot have rational periodic points of period 4 (Morton [38]), period 5 (Flynn–Poonen– +Schaefer [18]), or, assuming that the conclusions of the Birch and Swinnerton-Dyer conjecture +hold for a certain Jacobian variety, period 6 (Stoll [47]). In addition, a substantial amount +of empirical evidence supporting the assumption in Poonen’s theorem has been provided by +Hutz and Ingram [23] and Benedetto et al. [1]. However, it remains an open problem to +prove that this assumption is valid. Portraits for other families of quadratic maps over Q are +studied in the articles [3,6,32,33]. The present paper concerns the preperiodic portraits of +quadratic polynomials defined over quadratic fields, a topic previously explored in [10,12,13]. +1.2. Analogy with torsion points. A guiding principle that has proved fruitful in arith- +metic dynamics is to regard the set of preperiodic points of a map as being analogous to +the set of torsion points on an abelian variety. Thus, for instance, the well-known fact that +the set of K-rational torsion points on an abelian variety is finite is viewed as analogous to +a theorem of Northcott [41] stating that, for every rational function f over K of degree at +least 2, the set of K-rational preperiodic points of f is finite. +Motivated by this analogy, Morton and Silverman formulated the following dynamical +analogue of a standard uniform boundedness conjecture for abelian varieties. We state the +dynamical conjecture only in the case of endomorphisms of the projective line, although a +similar statement applies to arbitrary projective spaces1. +Conjecture 1.2 (Morton–Silverman [39]). For a number field K and morphism f : P1 +K → P1 +K +of degree greater than 1, the number of K-rational preperiodic points of f is bounded above +by a constant depending only on the degree of f and the absolute degree of K. +This dynamical uniform boundedness conjecture would imply, in particular, that there are +only finitely many isomorphism classes of portraits G(f, Q) as f ranges over all quadratic +polynomials with rational coefficients, since the number of vertices in such a portrait is +uniformly bounded. Theorem 1.1 can thus be seen as a refinement of the conjecture in this +case, as it provides a (conditionally) complete list of all possible portraits for the family of +quadratic polynomial maps. In the analogy with torsion points, Poonen’s list of portraits +corresponds to the list of abelian groups that can be realized as the torsion subgroup of an +elliptic curve over Q, the latter list being provided by a well-known theorem of Mazur [34]. +Similarly, the Morton–Silverman conjecture would imply that the portraits G(f, K), where +K is a quadratic field and f is a quadratic polynomial over K, fall into finitely many iso- +morphism classes. A conjecturally complete list of classes was first proposed in [13], and is +included here in Appendix B. The list is comprised of 46 portraits, and can be viewed as anal- +ogous to the list of 26 abelian groups, known by work of Kamienny [25] and Kenku–Momose +[26], that can arise as torsion subgroups of elliptic curves over quadratic fields. +1.3. Infinitely occurring portraits. Our primary objective in this paper is to determine, +under a suitable notion of equivalence of maps, which of the 46 graphs in [13] arise as the +preperiodic portrait of infinitely many inequivalent quadratic polynomials over quadratic +fields. In the context of elliptic curves, both over Q and over quadratic fields, the correspond- +ing question is well understood: every abelian group that arises as the torsion subgroup of +an elliptic curve can be realized as such by infinitely many non-isomorphic curves (see [24]). +1Interestingly, work of Fakhruddin [17] shows that the more general Morton–Silverman conjecture in fact +implies its analogue for abelian varieties. See also [45, §3.3], where Merel’s theorem for elliptic curves is +shown to follow from Conjecture 1.2 + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +3 +In order to state our questions more precisely, we begin by defining the appropriate equiv- +alence relation on maps. Two morphisms f, h : P1 +K → P1 +K are called linearly conjugate +over K if there exists an automorphism σ ∈ PGL2(K) such that +h = σ−1 ◦ f ◦ σ. +(Similarly, one can define linear conjugacy over any extension of K.) In that case, a simple +argument shows that the portraits G(f, K) and G(h, K) are isomorphic as directed graphs; +hence, the isomorphism class of G(f, K) is determined by the linear conjugacy class of f. +In the case of quadratic polynomials, it is well known that every such map f ∈ K[z] is +linearly conjugate to a unique map of the form +fc(z) := z2 + c, +where c ∈ K. Thus, in studying the portraits of quadratic polynomials we may restrict +attention to the one-parameter family of maps fc. +Returning to the question of portraits arising infinitely often, the case of quadratic poly- +nomials over Q was answered by Faber, who showed in addition that Poonen’s list in [43] +does not omit any such portrait. +Theorem 1.3 (Faber [16]). For a portrait P, the following are equivalent: +(i) There exist infinitely many c ∈ Q such that G(fc, Q) ∼= P. +(ii) P is isomorphic to one of the following graphs (using the labels from Appendix B): +∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1). +Motivated by Faber’s theorem, we now state the main questions to be addressed here. +Question 1.4. Among the 46 known isomorphism classes of portraits arising as G(fc, K), +with K a quadratic field and c ∈ K, which ones can be realized as such by infinitely many +algebraic numbers c? In addition, must every infinitely occurring portrait belong to one of +the 46 known isomorphism classes? +1.4. Main results. We define the following sets of portraits using labels as in Appendix B: +Γ0 := {∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1)}; +Γrat := {8(1, 1)a, 8(2)a, 8(4), 10(3, 1, 1), 10(3, 2)}; +Γquad := {8(1, 1)b, 8(2)b, 8(3), 10(2, 1, 1)a/b}; +Γ := Γ0 ∪ Γrat ∪ Γquad. +We provide two answers to Question 1.4 which differ in their level of specificity. The +simplest is Theorem 1.5, with Theorems 1.6 and 1.7 providing additional information in +terms of the above subsets of Γ. For an integer n ≥ 1, we define +Q(n) := {α ∈ Q : [Q(α) : Q] ≤ n}. +Theorem 1.5. For a portrait P, the following are equivalent: +(i) There exist infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field +K containing c. +(ii) P ∈ Γ. + +4 +JOHN R. DOYLE AND DAVID KRUMM +Theorem 1.5 can be refined in order to take into account certain subtleties illustrated by +the following example: We see from Theorem 1.3 that the portrait P = 4(2) is realized as +G(fc, Q) for infinitely many c ∈ Q. For each such c, the set of preperiodic points for fc is a +set of bounded height, and therefore fc has only finitely many preperiodic points of algebraic +degree 2 over Q. Hence, for each of the infinitely many c ∈ Q with G(fc, Q) ∼= P, we must +also have G(fc, K) ∼= P for all but finitely many quadratic fields K. +We therefore show that each of the portraits P ∈ Γ is realized infinitely often—even if +one excludes the infinitely many “trivial” realizations in the sense of the previous paragraph. +This is done in the next two theorems, which are stated separately in order to distinguish +between polynomials with rational coefficients and those with quadratic algebraic coefficients. +Theorem 1.6. For a portrait P, the following are equivalent: +(i) There exist infinitely many c ∈ Q such that G(fc, Q) ⊊ G(fc, K) ∼= P for some +quadratic field K. +(ii) P ∈ Γ ∖ {∅, 6(3)}. +Theorem 1.7. For a portrait P, the following are equivalent: +(i) There exist infinitely many c ∈ Q(2) ∖ Q such that G(fc, Q(c)) ∼= P. +(ii) P ∈ Γ0 ∪ Γquad. +Note that every portrait in Γ is covered by at least one of Theorems 1.6 and 1.7, since ∅ +and 6(3) are elements of Γ0. In particular, these two theorems together imply Theorem 1.5. +1.5. Quadratic points on dynamical modular curves. The proofs of our main results +rely heavily on the concept of a dynamical modular curve. To each of the 46 portraits from +[13], and more generally to any portrait that could potentially be realized as the preperi- +odic portrait of a quadratic polynomial over a number field, we associate an algebraic curve +parametrizing instances of the portrait as a preperiodic portrait G(fc, K). The curve cor- +responding to a portrait P will be denoted X1(P) by analogy with the classical modular +curves X1(N) parametrizing elliptic curves with a torsion point of order N. To avoid confu- +sion, the latter curve will henceforth be denoted Xell +1 (N). The details of the construction as +well as basic properties of dynamical modular curves are discussed in [11]. A more general +construction of dynamical moduli spaces appears in [15]. +The core of our analysis in this paper is a study of basic geometric invariants, such as +genus and gonality, of the curves X1(P). We then turn this geometric data into arithmetic +data using Faltings’ theorem on rational points on subvarieties of abelian varieties, via the +following result of Harris and Silverman: +Theorem 1.8 (Harris–Silverman [21, Cor. 3]). Let X be a smooth, irreducible, projective +curve of genus g ≥ 2 defined over a number field K. If X is neither hyperelliptic or bielliptic +over K, then X has only finitely many points that are quadratic over K. +In addition, we consider arithmetic questions regarding the fields of definition of quadratic +points on X1(P). In particular, if X1(P) has a point defined over a quadratic number field +K, what can be said about basic arithmetic invariants of K, such as discriminant and class +number? For the curves Xell +1 (N), arithmetic questions of this kind have been discussed by +several authors: Momose [35] shows that if K is the field of definition of a quadratic point +on Xell +1 (13), then the prime 2 splits in K, and 3 is unramified in K; Bosman et al. [5] show +that K must be a real quadratic field, an observation also made in [13]. In the case of the + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +5 +modular curves Xell +0 (N), Najman and Trbovi´c [40] prove arithmetic results of this type for +several values of N. +For the dynamical modular curves X1(P) we prove the following two theorems. Though +our methods can be applied to several portraits in the set Γ (namely, those for which the +corresponding modular curve is hyperelliptic), the portraits 8(4) and 10(3,1,1) are highlighted +here due to their significance in the context of elliptic curves, explained below. +By a quadratic point on an algebraic curve over a field k, we mean a point whose field +of definition is a quadratic extension of k. +Theorem 1.9. Let P denote the portrait 8(4). +(a) For every prime p and every residue class c ∈ Z/pZ, there exist infinitely many +squarefree integers d ∈ c such that X1(P) has a quadratic point defined over Q( +√ +d). +(b) There exist infinitely many imaginary quadratic fields K with class number divisible +by 10 such that X1(P) has a quadratic point defined over K. +As noted in [13], the above curve X1(P) is isomorphic to Xell +1 (16). Thus, Theorem 1.9 +provides new information about the collection of quadratic fields K such that there exists +an elliptic curve E/K with a K-rational torsion point of order 16. +Similarly, taking P = 10(3, 1, 1), the curve X1(P) is known to be isomorphic to Xell +1 (18). +The next theorem strengthens earlier results by Kenku–Momose [26] regarding the splitting +of rational primes in the fields of definition of quadratic points on this curve. +Theorem 1.10. Let P denote the portrait 10(3, 1, 1) and let K be the field of definition of +a quadratic point on X1(P). +(a) The prime 2 splits in K, and 3 is not inert in K. +(b) There exists an infinite and computable set of primes, denoted π, that is independent +of K, and such that every prime in π is unramified in K. +1.6. Points of higher degree. Though our primary focus here is on quadratic fields, we +make one observation concerning arbitrary number fields. The next result is a straightforward +consequence of a theorem of Frey [19] together with the main theorem of [14]. +Theorem 1.11. Fix a positive integer n. For any portrait P, let γ(P) denote the set of +algebraic numbers c ∈ Q(n) such that P ∼= G(fc, K) for some number field K satisfying +c ∈ K ⊂ Q(n). There are only finitely many portraits P such that γ(P) is infinite. +Note that Theorem 1.5 is a more refined version of Theorem 1.11 in the case n = 2. +1.7. Outline of the paper. In Section 2 we define the notion of a generic quadratic portrait +and discuss basic facts concerning dynamical modular curves associated to such portraits, +followed by general properties of algebraic curves in Section 3. +In Section 4, we apply geometric arguments to determine all generic quadratic portraits P +for which the curve X1(P) has infinitely many quadratic points. This proves, in particular, +the implication (i) ⇒ (ii) in Theorem 1.5; see Theorem 4.1 and the immediately preceding +discussion. +Section 5 addresses the issue that K-rational points on X1(P) correspond to instances +where G(fc, K) simply contains the portrait P; that is, we need not have an isomorphism +G(fc, K) ∼= P, and in fact, in many cases we do not. The section culminates with the proofs +of Theorems 1.6 and 1.7 in §5.3. + +6 +JOHN R. DOYLE AND DAVID KRUMM +Finally, Section 6 is devoted to arithmetic questions concerning the fields of definition of +quadratic points on the curves X1(P), and in particular to proving Theorems 1.9 and 1.10. +Acknowledgements. We thank Joe Silverman for helpful comments, and especially for a +suggestion that led to the more refined statements in Theorems 1.6 and 1.7. The first author +was partially supported by NSF grant DMS-2112697. +2. Dynamical modular curves +2.1. Dynatomic polynomials. If f is a polynomial with coefficients in a field K and α ∈ K +is a point of exact period n for f, then α is a root of the polynomial f n(z)−z. However, the +roots of f n(z) − z may have period strictly dividing n, and indeed there is a factorization +f n(z) − z = +� +d|n +Φd,f(z), +where (generically) the roots of Φd,f have exact period d for f. M¨obius inversion yields +Φn,f(z) = +� +d|n +(f d(z) − z)µ(n/d), +where µ denotes the M¨obius function. We call Φn,f the nth dynatomic polynomial of f. +More generally, for m, n ≥ 1 we define +Φm,n,f(z) := +Φn,f(f m(z)) +Φn,f(f m−1(z)). +Then Φm,n,f is a polynomial whose roots are (again, generically) points of preperiod m and +eventual period n for f. (That Φn,f and Φm,n,f are indeed polynomials is proven in [22,45].) +Since we are specifically interested in the family fc(z) = z2 + c, we write +Φn(c, z) := Φn,fc(z) +and +Φm,n(c, z) := Φm,n,fc(z). +Then Φn (resp., Φm,n) is a polynomial in Z[c, z], and the vanishing locus defines an affine +curve Y1(n) (resp., Y1(m, n)), which we refer to as a dynatomic curve. Thus, for example, +if α has period n for fc, then (c, α) is a point on the dynatomic curve Y1(n). We denote by +X1(·) the normalization of the projective closure of the affine curve Y1(·), and we also refer +to X1(·) as a dynatomic curve. +2.2. Dynamical modular curves associated to portraits. The dynamical properties +of quadratic polynomial maps impose certain restrictions on those portraits that may be +realized as G(f, K) for some number field K and a quadratic polynomial f ∈ K[z]. First, +no point may have more than two preimages under f. Also, for each positive n ∈ Z, the nth +dynatomic polynomial for a quadratic polynomial f has degree +(2.1) +D(n) := deg Φn,f(z) = +� +d|n +µ(n/d)2d. +Thus, a quadratic polynomial has at most D(n) points of period n, partitioned into at most +R(n) := D(n)/n cycles of length n. With these restrictions in mind, we make the following +definition: + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +7 +• +• +• +• +Figure 1. A generic quadratic portrait +Definition 2.1. A quadratic portrait is a portrait P satisfying the following properties: +(a) Every vertex of P has in-degree at most 2. +(b) For each n ≥ 1, the number of n-cycles in P is at most +R(n) := 1 +n +� +d|n +µ(n/d)2d. +For any number field K and quadratic polynomial f ∈ K[z], the portrait G(f, K) is +quadratic. However, for most quadratic polynomials (in a sense that can be made precise), +we can say more about the structure of the set of K-rational preperiodic points. For the +model fc(z) = z2 + c, if α is a preperiodic point for fc, then −α is also preperiodic, since +both are preimages of f(α). Thus, a preperiodic point typically has either no K-rational +preimages or exactly two K-rational preimages. The exception to this rule occurs when +α = 0 is a preperiodic point, in which case exactly one preperiodic point (namely, c = fc(0)) +has a single K-rational preimage. +Along the same lines, if a polynomial fc has a K-rational fixed point β, then β is a root of +the quadratic polynomial fc(z)−z = z2−z+c; thus, unless we have disc(fc(z)−z) = 1−4c = 0 +(i.e., c = 1/4), there is a second fixed point β′, necessarily defined over K. +With these observations in mind, we make the following definition. +Definition 2.2. A generic quadratic portrait is a quadratic portrait P with the following +additional properties: +(a) The in-degree of any vertex of P is equal to 0 or 2. +(b) If P has a fixed point, then P has exactly two fixed points. +Remark 2.3. We will sometimes refer to the results of [11], in which the term “strongly +admissible” is used instead of “generic quadratic.” +Given a quadratic portrait P, there is a dynamical modular curve Y1(P), defined over +Q, whose K-points—for any extension K/Q—correspond to tuples (c, z1, . . . , zn) such that +z1, . . . , zn are preperiodic points forming a subportrait of G(fc, K) isomorphic to P. If Q +is the point on Y1(P) corresponding to such a tuple, then the field of definition of Q is +Q(c, z1, . . . , zn). We denote by X1(P) the smooth projective curve birational to Y1(P). +A formal treatment of dynamical modular curves appears in [11], where the curves are de- +fined only for generic quadratic portraits. We lose no generality in making such a restriction: +Given any quadratic portrait P, there is a unique portrait P′ that is minimal among generic +quadratic portraits containing P as a subportrait. In the language of [11], P′ is the generic +quadratic portrait generated by the vertices of P. It follows from the results of [11, §2] that +X1(P) ∼= X1(P′), so we may as well assume that P is generic quadratic. +Rather than formally defining X1(P) (we refer the interested reader to [11] or, for a +different approach in a more general setting, [15]), we give an example. +Example 2.4. Consider the generic quadratic portrait P appearing in Figure 1. One could +construct a curve birational to X1(P) simply by giving one equation for each relation coming + +8 +JOHN R. DOYLE AND DAVID KRUMM +from an edge in P: if we label the vertices 1, 2, 3, 4 from left to right, we have +z2 +1 + c = z2, +z2 +2 + c = z3, +z2 +3 + c = z2, +z2 +4 + c = z3. +(2.2) +Note that we must also impose certain Zariski open conditions of the form zi ̸= zj to remove +extraneous components; for example, there is a full component of the curve defined by (2.2) +on which z1, z2, z3, and z4 are all equal. It is this approach that is taken in [15]. +An alternative approach, which is particular to quadratic polynomials, is to note that any +generic quadratic portrait containing a point of period 2 must necessarily contain P. Thus, +another (affine) model for X1(P) is the plane curve defined by the vanishing of +Φ2(c, z) = (z2 + c)2 + c − z +z2 + c − z += z2 + z + c + 1. +In other words, X1(P) is isomorphic to the dynatomic curve X1(2). This second model has +the advantage of being defined in a lower-dimensional affine space (A2, rather than A5), and +it is this second approach which is described in detail in [11]. +We conclude this example by pointing out that X1(P) ∼= X1(2) also has “degenerate” +points where two or more of the vertices of the portrait P collapse. +For example, the +equation Φ2(c, z) = 0 has the solution (c, z) = +� +− 3 +4, − 1 +2 +� +despite the fact that − 1 +2 is a fixed +point for f−3/4. However, for a given portrait P, there are only finitely many such degenerate +points on X1(P). +Before summarizing the required properties of dynamical modular curves, we recall the +following terminology: +Definition 2.5. Let X be a smooth, irreducible projective curve defined over a field k. +The k-gonality of X, denoted gonk(X), is the minimal degree of a nonconstant morphism +X → P1 defined over k. +Proposition 2.6. Let P be a generic quadratic portrait, and let k be any field of character- +istic 0. +(a) The curve X1(P) is irreducible over k. +(b) If P′ is a generic quadratic portrait properly contained in P, then there is a finite +morphism πP,P′ : X1(P) → X1(P′) of degree at least 2 defined over k. +(c) Given any ordering P1, P2, . . . of all generic quadratic portraits, the k-gonalities of +the curves X1(Pi) tend to ∞. +Proof. Parts (a) and (b) are proven in [11, Thm. 1.7] and [11, Prop. 3.3], respectively. Note +that the morphism πP,P′ is obtained simply by forgetting the preperiodic points correspond- +ing to vertices of P ∖ P′, hence is defined over the base field k. +Statement (c) is a slight generalization of, but follows directly from, [14, Thm. 1.1(b)], +which says that as m + n → ∞, the gonalities of the curves X1(m, n) tend to ∞. Given a +bound B, there are only finitely many quadratic portraits P such that every vertex v of P +has preperiod m and eventual period n satisfying m + n ≤ B. In other words, if for every +generic quadratic portrait P we choose a vertex vP with preperiod mP and eventual period +nP maximizing the sum mP + nP, we must have mP + nP → ∞ as P ranges over all generic +quadratic portraits in any order. Since there is a nonconstant morphism from X1(P) to +X1(mP, nP) (e.g., by part (b)), we have gonk(X1(P)) ≥ gonk(X1(mP, nP)), and the latter +expression tends to ∞. +□ + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +9 +Proof of Theorem 1.11. Fix n ≥ 1 and a portrait P, and suppose there are infinitely many +c ∈ Q(n) such that G(fc, K) ∼= P for some degree-n number field K containing c. Then the +dynamical modular curve X1(P) has infinitely many points of degree at most n. It follows +from [19, Prop. 2] (cf. [8, Thm. 5]) that X1(P) must have gonality at most 2n, hence there +are only finitely many such portraits P by part (c) of Proposition 2.6. +□ +3. Some useful properties of algebraic curves +In this section, we collect a few facts about algebraic curves that will be used throughout +the rest of the paper. +First, we provide a statement that follows from Hilbert’s irreducibility theorem; see [44, +§3.4] and [30, §9.2] for details. +Proposition 3.1. Let K be a number field, let X be a curve defined over K, and let ϕ : +X → P1 be a dominant morphism of degree d ≥ 2 defined over K. Then the set +T := +� +P ∈ P1(K) : [K(Q) : K] < d for some Q ∈ ϕ−1(P) +� +is a thin subset of P1(K). +Remark 3.2. Thin subsets T ⊂ P1(K) have density 0, in the sense that +lim +N→∞ +���{P ∈ T : h(P) ≤ N} +��� +���{P ∈ P1(K) : h(P) ≤ N} +��� += 0, +where h is the na¨ıve Weil height on P1(Q). In particular, for any maximal ideal p ∈ Spec OK +and any mod-p residue class c in P1(K), the set c \ T is infinite. +Given an elliptic curve E with Weierstrass equation y2 = f(x), where f ∈ K[x] is square- +free of degree 3, it is easy to construct infinitely many quadratic points on E: For “most” +x ∈ K, the point (x, y) = (x, +� +f(x)) is quadratic over K. More precisely, since f is not a +square in K[x], it follows from Hilbert irreducibility that f(x0) is a nonsquare in K for all +x0 outside a thin subset of K. The following result, proven in [13, Lem. 2.2], gives a useful +characterization of quadratic points (x, y) with x /∈ Q: +Lemma 3.3. Let E/K be an elliptic curve defined by an equation of the form +y2 = ax3 + bx2 + cx + d, +where a, b, c, d ∈ K and a ̸= 0. Suppose (x, y) ∈ E(K) is a quadratic point with x /∈ K. +Then there exist (x0, y0) ∈ E(K) and t ∈ k such that y = y0 + t(x − x0) and +x2 + ax0 − t2 + b +a +x + ax2 +0 + t2x0 + bx0 − 2y0t + c +a += 0. +By Theorem 1.8, a curve with infinitely many quadratic points must admit a degree-2 +morphism to either P1 or an elliptic curve, hence must have gonality at most 4. Thus, to +prove that a curve has finitely many quadratic points, it suffices to show that the gonality +of the curve is greater than 4. The following inequality is a standard tool for finding lower +bounds for gonalities. + +10 +JOHN R. DOYLE AND DAVID KRUMM +Proposition 3.4 (Castelnuovo–Severi inequality [46, Thm. 3.11.3]). Let Y , Y1, and Y2 be +curves of genera gY , g1, and g2, respectively. Suppose we have maps ϕ1 : Y → Y1 and +ϕ2 : Y → Y2 of degrees d1 and d2, and suppose further that there is not an intermediate +curve Z and a map ψ : Y → Z of degree greater than 2 such that both ϕ1 and ϕ2 factor +through ψ. Then +(3.1) +gY ≤ d1g1 + d2g2 + (d1 − 1)(d2 − 1). +4. Dynamical modular curves with infinitely many quadratic points +The purpose of this section is to prove one direction of Theorem 1.5, namely that if there +are infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field K containing +c, then P ∈ Γ. Since any such realization of P as G(fc, K) yields a quadratic point on the +dynamical modular curve Y1(P), it suffices to prove the following: +Theorem 4.1. Let P be a generic quadratic portrait. +Then X1(P) has infinitely many +quadratic points if and only if P ∈ Γ. +Remark 4.2. If we just assume that P is a quadratic portrait (i.e., not necessarily generic), +then X1(P) has infinitely many quadratic points if and only if P is a subportrait of some +portrait in Γ. This follows from Theorem 4.1 as well as the fact that for any quadratic +portrait P, if we let P′ be the minimal generic quadratic portrait containing P, then X1(P) +and X1(P′) are isomorphic over Q. (See the discussion preceding Example 2.4.) +One direction of Theorem 4.1 is straightforward: For every portrait P ∈ Γ, the curve +X1(P) is described in at least one of the articles [38,43,48]. All the curves in those articles +have genus at most 2 and at least one rational point, hence have infinitely many quadratic +points. Thus, we must show that if P is generic quadratic but not in Γ, then X1(P) has only +finitely many quadratic points. +To help organize the arguments in the rest of this section, we introduce some terminology: +Definition 4.3. The cycle structure of a portrait P is the nonincreasing sequence of cycle +lengths appearing in P. Note that the empty portrait has cycle structure ( ). +If K is a quadratic field and c ∈ K, then the cycle structure of G(fc, K) may contain the +integer 1 at most twice and each of the integers 2, 3, and 4 at most once; for periods 1 and +2 this follows from the fact that a quadratic polynomial can have at most two fixed points +and at most one 2-cycle, and for periods 3 and 4 this comes from [10, Cor. 4.16]. More +precisely, the results of [10] imply that the “period at most 4” portion of the cycle structure +of G(fc, K) must be (4,1,1), (4,2), or one of the following: +(4.1) +( ), (1,1), (2), (3), (4), (2,1,1), (3,1,1), (3,2). +Moreover, it follows from [13, Cor. 3.48] (resp., [10, Thm. 4.21]) that no portrait with +both a 4-cycle and a 1-cycle (resp., 4-cycle and a 2-cycle) may be realized infinitely often as +G(fc, K) for K a quadratic field and c ∈ K. For our purposes, therefore, we may exclude +the cycle structures (4,1,1) and (4,2) from consideration. +By enumerating generic quadratic portraits with few vertices, one can verify that if P is a +generic quadratic portrait which is not in Γ, then P has a cycle of length n ≥ 5 or P properly +contains a portrait in Γ1 or Γ2. We handle these two possibilities separately, showing in each +case that the dynamical modular curve X1(P) has only finitely many quadratic points. + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +11 +4.1. Points of period n ≥ 5. If P is a generic portrait with a cycle of length n, then there +is a dominant morphism X1(P) → X1(n) defined over Q. In particular, every quadratic +point on X1(P) maps to a rational or quadratic point on X1(n), so we need only show that +if n ≥ 5, then X1(n) has only finitely many points defined over quadratic fields; this is the +content of Proposition 4.6. +For n ≥ 1, the cyclic group Cn acts on X1(n) as follows: Given a point (c, z) ∈ X1(n), we +also have σn(c, z) := (c, fc(z)) ∈ X1(n), so σ defines an order-n automorphism of X1(n). We +denote by πn : X1(n) → X0(n) the quotient of X1(n) by this cyclic group action. (The curve +X0(n) parametrizes maps fc together with a marked cycle of length n.) Let c1,n and c0,n +denote the maps from X1(n) and X0(n), respectively, to the c-line; note that c1,n = c0,n ◦ πn. +For a curve X, we will denote by gX its genus; for simplicity, for each n ≥ 1 we will write +g1,n and g0,n for the genera of X1(n) and X0(n), respectively. Finally, recall that we denote +by D(n) the degree (in z) of the polynomial Φn(c, z); a formula for D(n) is given in (2.1), +and using that formula one can show that +(4.2) +2n−1 ≤ D(n) ≤ 2n, +with equality on the right if and only if n = 1 and on the left if and only if n = 2. +Lemma 4.4. For all n ≥ 5, we have g0,n ≥ 2. +Proof. In [37, Thm. 13], Morton gives an explicit formula for g0,n as well as the lower bound +g0,n ≥ 3 +2 + +�1 +4 − 1 +n +� +2n − (n + 1)2n/2−1. +Rewriting the right-hand expression and using the assumption that n ≥ 5, we have +g0,n ≥ 3 +2 + 2n/2−1 +��1 +4 − 1 +5 +� +2n/2+1 − (n + 1) +� += 3 +2 + 2n/2−1 +� 1 +102n/2 − (n + 1) +� +. +The expression +� 1 +102n/2 − (n + 1) +� +is positive for all n ≥ 15, so for all such n we have +g0,n > 3/2, hence g0,n ≥ 2. Finally, using the explicit formula for g0,n given by Morton, +we can exactly compute g0,n for all 1 ≤ n ≤ 14, and we find that g0,n < 2 if and only if +1 ≤ n ≤ 4, in which case g0,n = 0. +□ +Lemma 4.5. Let n ≥ 6, and let Rn be the ramification divisor of πn. Then deg Rn > 4n. +Proof. It suffices to replace Rn with R0 +n, the restriction of the ramification divisor to points +that do not map to ∞ under c1,n. The ramification divisors of the maps c1,n and c0,n are +explicitly computed by Morton in [37, Thms. 11, 13], from which it follows that +(4.3) +deg R0 +n = 1 +2 +� +d|n +d 2m−2ϕ(n/m)n +2 +(by (4.2)) +≥ 2 +√n−3ϕ(n/m)n. +For all n ≥ 25, we have 2 +√n−3 ≥ 4, thus deg R0 +n > 4n. An explicit computation of deg R0 +n +for all 6 ≤ n ≤ 25 (using (4.3)) completes the proof. +□ +Proposition 4.6. Let n ≥ 5. Then X1(n) has only finitely many quadratic points. +Remark 4.7. The fact that X1(n) has only finitely many quadratic points for sufficiently +large n follows from Theorem 1.8, together with [14, Thm. +1.1], which states that the +gonality of X1(n) tends to infinity with n. For our purposes, however, we need the more +precise statement of Proposition 4.6. +Proof of Proposition 4.6. By Theorem 1.8, it suffices to show that for n ≥ 5, the curve X1(n) +does not admit a degree-2 map to a curve of genus at most 1. +Let ϕ : X1(n) → C be a dominant morphism, where C is a curve of genus gC ≤ 1. We +claim that d := deg ϕ > 2. +First, suppose n ≥ 6. +In order to apply the Castelnuovo–Severi inequality (Proposi- +tion 3.4), we consider two cases. +Case 1: Suppose there is a curve Y such that both πn : X1(n) → X0(n) and ϕ : X1(n) → +C factor through a map ψ : X1(n) → Y of degree at least 2. Since Y covers X0(n), we have +gY ≥ 2 > gC by Lemma 4.4, hence the map Y → C has degree at least 2. Therefore, d ≥ 4. +Case 2: Now suppose that there is no such curve Y , so that we can apply the Castelnuovo– +Severi inequality to the maps ϕ and πn to get +g1,n ≤ ng0,n + dgC + (n − 1)(d − 1). +Rewriting this, we have +g1,n − ng0,n + n − 1 ≤ d(gC + n − 1). +By the Riemann–Hurwitz formula, the left hand side is equal to half the degree of the +ramification divisor Rn, so by Lemma 4.5 we have +d(gC + n − 1) > 2n. +Since we assumed gC ≤ 1, this implies that d > 2. +Finally, we consider n = 5. A calculation in Magma [4] shows that the map X1(5) → P1 +given by Φ2(c, z) = z2 + z + c + 1 has degree 7. If ϕ factors through Φ2, then d ≥ 7. If +not, then ϕ and Φ2 do not simultaneously factor through a nontrivial intermediate map +X1(5) → Y , since Φ2 has prime degree. By the Castelnuovo–Severi inequality, we have +g1,5 ≤ dgC + 6(d − 1). +The curve X1(5) has genus g1,5 = 14, and we assumed gC ≤ 1, so it follows that d > 2. +□ + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +13 +• +• +• +• +• +• +• +• +• +• +• +Figure 2. The portrait 10(4) +Remark 4.8. The fact that Φ2 has low degree on X1(5) seems related to the fact that, as +polynomials in Q[c, z], the dynatomic polynomials Φ2 and Φ5 have no common zeros (c, z). +In particular, all zeros and poles of Φ2 on X1(5) lie above ∞, which restricts the possible +number of such points. +4.2. Generic quadratic portraits properly containing the portraits in Γ1 and Γ2. +By enumerating portraits with few vertices, one finds that any generic quadratic portrait P +that has its cycle structure listed in (4.1), but which is not contained in Γ, must properly +contain a portrait from Γ1 or Γ2, and moreover, P must have a subportrait isomorphic to +one of the following portraits: +(4.4) +10(1,1)a/b, 10(2), 10(3)a/b, 10(4), 12(2,1,1)a/b, or Gn for some 1 ≤ n ≤ 10. +All portraits listed above appear in Appendix B except 10(4), which is the label we give to +the subportrait of 12(4) shown in Figure 2. +Proposition 4.9. For each of the portraits P appearing in (4.4), the curve X1(P) has only +finitely many quadratic points. +The cases P = 10(1, 1)b and P = 10(2) form the majority of the proof of Proposition 4.9. +We include only the proof for 10(1, 1)b, as the argument for 10(2) is very similar. +Lemma 4.10. Let C ⊂ Spec Q[x, z] be the curve of genus 5 defined by the equation +� +z2 − 2(x2 + 1) +�2 = 2(x2 − 1)2(x3 + x2 − x + 1). +Let (c, p) ∈ A2(K) be such that p has preperiod 4 and eventual period 1 for fc. Then there +exists a point (x, z) ∈ C(K) such that c = −2(x2 + 1)/(x2 − 1)2. +Proof. Let q = fc(p) = p2 + c, so that q has preperiod 3 (and still eventual period 1). A +calculation in [43, p. 22] shows that there is an element x ∈ K ∖ {±1} such that +c = −2(x2 + 1) +(x2 − 1)2 +and q2 = 2(x3 + x2 − x + 1) +(x2 − 1)2 +. +Hence we have +2(x3 + x2 − x + 1) = q2(x2 − 1)2 = (p2 + c)2(x2 − 1)2 = +�p2(x2 − 1)2 − 2(x2 + 1) +x2 − 1 +�2 +. +Letting z = p(x2 − 1) we obtain +� +z2 − 2(x2 + 1) +�2 = (x2 − 1)2 · 2(x3 + x2 − x + 1) +with x, z ∈ K. Thus (x, z) ∈ C(K). +□ + +14 +JOHN R. DOYLE AND DAVID KRUMM +Proposition 4.11. For the portraits P = 10(1, 1)b and P = 10(2), the set of quadratic +points on X1(P) is finite. +Proof. As mentioned above, we only give a proof for P = 10(1, 1)b. By Lemma 4.10, it +suffices to show that the curve C has only finitely many quadratic points. The latter curve +admits a dominant map to the elliptic curve with Weierstrass equation w2 = 2(x3+x2−x+1), +which is the modular curve Xell +1 (11). Explicitly, a natural map ϕ : C → Xell +1 (11) is given by +ϕ(x, z) = +� +x, z2 − 2(x2 + 1) +x2 − 1 +� +. +The curve Xell +1 (11) has exactly four affine rational points, namely, (±1, ±2). Suppose +that (x, z) is a quadratic point on C with field of definition K. Then x /∈ {±1}, so the +point ϕ(x, z) cannot be a rational point on X1(11). Thus ϕ(x, z) is a quadratic point, and +K = Q(ϕ(x, z)). Letting +(4.5) +w = z2 − 2(x2 + 1) +x2 − 1 +, +we therefore have w2 = 2(x3 + x2 − x + 1) and K = Q(x, w). We now consider two cases. +Suppose first that x ∈ Q. Then K = Q(w), and by (4.5) we have z2 = 2(x2+1)+(x2−1)w. +Applying the norm map NK/Q to this equation we obtain +y2 = 4(x2 + 1)2 − 2(x2 − 1)2(x3 + x2 − x + 1), +where y = NK/Q(z). +The above equation defines a hyperelliptic curve of genus 3, and +therefore has only finitely many rational solutions. We conclude that C has only finitely +many quadratic points with rational x-coordinate. +Now suppose that x /∈ Q, so that K = Q(x). By Lemma 3.3 applied to the equation +w2 = 2(x3 + x2 − x + 1), there is a rational number t and a point (x0, w0) ∈ {(±1, ±2)} such +that w = w0 + t(x − x0) and +(4.6) +x2 + 2x0 − t2 + 2 +2 +x + 2x2 +0 + t2x0 + 2x0 − 2w0t − 2 +2 += 0. +For each point (x0, w0) ∈ {(±1, ±2)} we consider the relation +(4.7) +z2 = 2(x2 + 1) + (x2 − 1)(w0 + t(x − x0)). +Using (4.6) we express the right-hand side of (4.7) as a linear combination of 1 and x. +Applying the norm map NK/Q and letting u = 2 · NK/Q(z), we obtain a relation of the +form u2 = g(t), where g is a polynomial of degree 7 with integral coefficients and nonzero +discriminant. Each of the resulting four equations u2 = g(t) defines a hyperelliptic curve of +genus 3, and therefore has only finitely many rational solutions. Since t has only finitely +many possible values, (4.6) implies the same for x. Therefore C has finitely quadratic points +with quadratic x-coordinate. +□ +Proof of Proposition 4.9. The proposition has already been proven in [13] and [10] for all of +the portraits except 10(1, 1)b, 10(2), and 10(3)a/b. Moreover, Proposition 4.11 shows that +the statement is true for the portraits 10(1,1)b and 10(2), so all that remains is to show that +X1(P) has only finitely many quadratic points when P = 10(3)a or P = 10(3)b. +Each of the curves X1(P) with P = 10(3)a/b has genus 9, and each admits a degree-2 +map ϕ to the genus-2 curve X1(P′), where P′ = 8(3). Now suppose we have a degree-d map +ψ : X1(P) → C, where C is a curve of genus gC ≤ 1. Then, by Proposition 3.4, either ψ + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +15 +factors through ϕ, in which case deg ψ > deg ϕ = 2, or the Castelnuovo–Severi inequality +(3.1) applies to ϕ and ψ, in which case we have +4 + dgC + (d − 1) ≥ 9, hence d ≥ +6 +gC + 1 ≥ 3. +It follows that X1(P) is not hyperelliptic or bielliptic, hence X1(P) has only finitely many +quadratic points by Theorem 1.8. +□ +4.3. Proof of Theorem 4.1. We now combine Propositions 4.6 and 4.9 to complete the +proof of Theorem 4.1, which in turn proves one direction of Theorem 1.5. +Proof of Theorem 4.1. As mentioned previously, the fact that X1(P) has infinitely many +quadratic points for each P ∈ Γ follows from the work of Walde–Russo [48] and Poonen +[43]. Now suppose P is a generic quadratic portrait such that X1(P) has infinitely many +quadratic points. +Proposition 4.6 asserts that P cannot have a cycle of length n ≥ 5; +combining this with the paragraph following Definition 4.3, the cycle structure of P must be +one of those appearing in (4.1). By simply enumerating all small generic quadratic portraits +with the allowable cycle structures, one finds that if P is not contained in Γ, then P has +a subportrait isomorphic to one of the portraits listed in (4.4), hence there is a dominant +morphism X1(P) → X1(P′) for some P′ in that list. Proposition 4.9 shows that each such +X1(P′) has only finitely many quadratic points, hence X1(P) does as well. +□ +5. Preperiodic portraits realized infinitely often over quadratic fields +In this section, we show that if P ∈ Γ, then there are infinitely many c ∈ Q such that +G(fc, K) ∼= P for some quadratic field K. We also determine for which portraits P the same +is true for infinitely many c ∈ Q(2) ∖ Q. +It follows from Theorem 4.1 that Γ is precisely the set of generic quadratic portraits that +can be realized infinitely often as a subportrait of G(fc, K); the difficulty is in proving that +we infinitely often have equality. This step requires two main tools: The first is Hilbert +irreducibility, and the second is a dynamical result giving an upper bound for the lengths of +periodic cycles of maps over number fields that depends only on the primes of bad reduction +of those maps; see, for example, [42,45,49]. +Proposition 5.1. Let K be a number field, and let p ∈ Spec OK be a prime ideal of norm q. +There exists a bound B := B(q) such that if c ∈ K and vp(c) ≥ 0, then fc has no K-rational +points of period larger than B. +We will also repeatedly use the observation that given any portrait P and any bound C, +there are only finitely many generic portraits P′ that are minimal (relative to inclusion) +among those generic portraits containing P and have no cycles of length larger than C. This +is more or less due to the fact that there are only finitely many generic quadratic portraits +with a given number of vertices. +5.1. Rational c-values. For any c ∈ Q, there are infinitely many quadratic fields K for +which G(fc, K) ∼= G(fc, Q). +This is a consequence of Northcott’s theorem: The set of +preperiodic points for fc has bounded height, hence there are only finitely many preperiodic +points which are quadratic over Q, and therefore only finitely many quadratic fields over +which fc gains new preperiodic points. + +16 +JOHN R. DOYLE AND DAVID KRUMM +Table +1. For +each +pair +(P, P′), +there +is +a +degree-2 +morphism +X1(P) → X1(P′) defined over Q. +P +4(1,1) +4(2) +6(1,1) +6(2) +8(2,1,1) +P′ +∅ +∅ +4(1,1) +4(2) +4(1,1) or 4(2) +In particular, if a portrait P is realized as G(fc, Q) for infinitely many c ∈ Q, then P must +also be realized as G(fc, K) for infinitely many c ∈ Q and, for each such c, infinitely many +quadratic fields K. The portraits realized infinitely often over Q are precisely the portraits +in Γ0; this is the main result of [16]. +A more interesting problem, then, is to determine the set of portraits P for which there +are infinitely many c ∈ Q with G(fc, Q) ⊊ P but G(fc, K) ∼= P for some quadratic field K. +Proposition 5.2. Let P ∈ Γ. Then there exist infinitely many c ∈ Q such that G(fc, K) ∼= P +for some quadratic field K. Moreover, if P ∈ Γ ∖ {∅, 6(3)}, the infinitely many c ∈ Q may +be chosen so that +G(fc, Q) ⊊ G(fc, K) ∼= P. +Remark 5.3. The portraits ∅ and 6(3) are genuine exceptions to the second statement, as +asserted in Theorem 1.6 and proven in §5.3. +Proof of Proposition 5.2. It follows from the discussion preceding the statement of Proposi- +tion 5.2 that for both P = ∅ and P = 6(3), which are elements of Γ0, there are infinitely +many c ∈ Q such that G(fc, K) ∼= P for some quadratic field K. We henceforth assume +P ∈ Γ ∖ {∅, 6(3)} and prove the stronger statement that there are infinitely many c ∈ Q +such that G(fc, Q) ⊊ G(fc, K) ∼= P for some quadratic field K. +There is a model for X1(P) of the form y2 = F(x), with F(x) ∈ Q[x] nonconstant and +squarefree, such that the morphism c : X1(P) → P1 factors through x : X1(P) → P1. If +X1(P) has genus 0, this is because there is a proper (generic quadratic) subportrait P′ ⊊ P +for which the natural morphism +πP,P′ : X1(P) −→ X1(P′) +described in Proposition 2.6(b) has degree exactly 2; see Table 1 for the list of such pairs +(P, P′). For the curves of genus 1 or 2, explicit models are given in Appendix A. +Now fix a portrait P ∈ Γ ∖ {∅, 6(3)}, and let y2 = F(x) be the model for X1(P) described +in the previous paragraph. Choose any value of x0 ∈ Q, and choose a prime p ∈ Spec Z +of good reduction for c : X1(P) → P1 such that vp(c(x0)) ≥ 0. Let B = B(p2) be the +bound from Proposition 5.1, let P1, . . . , Pn be the generic quadratic portraits that properly +contain P and that have no cycles of length larger than B, and for each i = 1, . . . , n let +πi := πPi,P : X1(Pi) → X1(P) be the natural projection morphism from Proposition 2.6(b). +By Hilbert’s Irreducibility Theorem, the sets +{x ∈ Q : +� +F(x) ∈ Q} +and, for each i = 1, . . . , n, +� +x ∈ Q : +� +Q +� +π−1 +i +� +x, +� +F(x) +�� +: Q +� +≤ 2 +� +, +are thin subsets of P1(Q). Since the residue class +[x0]p := {x ∈ P1(Q) : x ≡ x0 (mod p)} + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +17 +is not thin, there are infinitely many x ∈ [x0]p such that K := Q +�� +F(x) +� +is a quadratic +field and +� +x, +� +F(x) +� +∈ X1(P)(K) does not lift to a K-rational point on X1(Pi) for any +i = 1, . . . , n. +Now let c = c(x) for any of the infinitely many x from the previous paragraph. Excluding +at most finitely many x ∈ [x0]p, we may assume that G(fc, K) is a generic quadratic portrait. +Since c lifts to a quadratic point Q on X1(P), we have +G(fc, Q) ⊊ P ⊆ G(fc, K). +On the other hand, since c does not lift to a quadratic point on X1(Pi) for any i = 1, . . . , n, +the portrait G(fc, K) is either isomorphic to P or contains a cycle of length greater than B. +Finally, since vp(c) ≥ 0, G(fc, K) has no cycles of length greater than B, and thus we have +G(fc, Q) ⊊ G(fc, K) ∼= P. +□ +5.2. Quadratic c-values. The purpose of this section is to determine which portraits P +may be realized infinitely often as G(fc, Q(c)) for some quadratic algebraic number c. We +begin with the simplest case, namely, where X1(P) has genus 0. +Proposition 5.4. Suppose P ∈ Γ0, and let K/Q be a quadratic field. +Then there are +infinitely many c ∈ K ∖ Q such that G(fc, K) ∼= P. +Proof. Let X := X1(P) ∼=Q P1. Choose an inert prime p ∈ Spec Z such that c : X → P1 +has good reduction at p and such that p > D := deg(c : X → P1). Let p ∈ Spec OK be the +unique prime lying above p. The good reduction condition implies that the mod-p reduction +�c of the map c : X → P1 has degree D, and the condition that D < p implies that �c cannot +map all of P1(Fp2) to P1(Fp). In other words, we may choose a point P0 ∈ X(K) such that +� +c(P0) = �c(� +P0) ∈ P1(Fp2) ∖ P1(Fp). Note that since ∞ ∈ P1(Fp) and � +c(P0) /∈ P1(Fp), we must +have vp(c(P0)) ≥ 0. Then, for any P in the residue class [P0]p, c(P) must be in K ∖ Q, +and vp(c(P)) ≥ 0. In particular, for all P ∈ [P0]p, fc(P) has no K-rational points of period +greater than B = B(p2), the bound from Proposition 5.1. +Let P1, . . . , Pn be the complete list of generic quadratic portraits that minimally contain +P and which have no cycles of length larger than B, and for each i = 1, . . . , n let +πi := πPi,P : X1(Pi) −→ P1 +be the morphism from Proposition 2.6(b). If P ∈ [P0]P and G(fc(P), K) properly contains +P, then G(fc(P), K) must contain one of the portraits Pi. By Hilbert irreducibility, the set +[P0]p ∖ +n� +i=1 +πi +� +X1(Pi)(K) +� +is infinite. Thus, there are infinitely many c ∈ K ∖ Q such that G(fc, K) ∼= P. +□ +We now consider the portraits P ∈ Γ ∖ Γ0; that is, the portraits for which X1(P) has +genus 1 or 2. We begin with a useful consequence of Theorem 4.1. +Corollary 5.5. Let P be a generic quadratic portrait such that X1(P) has positive genus. +If P′ is any generic quadratic portrait properly containing P, then X1(P′) has finitely many +quadratic points. + +18 +JOHN R. DOYLE AND DAVID KRUMM +Proof. If X1(P′) has infinitely many quadratic points, then so does X1(P), and therefore +both P and P′ are elements of Γ. By simply inspecting the portraits in Γ, the only way we +can have P, P′ ∈ Γ and P ⊊ P′ is if P ∈ Γ0; that is, if X1(P) has genus 0. +□ +Combining Corollary 5.5 and Proposition 5.1 gives us the following sufficient condition for +P to be realized infinitely often as G(fc, K) over quadratic fields K. For an algebraic curve +X defined over a field k, we denote by X(k, 2) the set of all points on X of degree at most +2 over k. For a prime p ∈ Spec Z and an element α ∈ Q, the phrase “vp(α) ≥ 0” should be +read to mean “there exists some extension p ∈ Spec OQ(α) of p such that vp(α) ≥ 0.” +Lemma 5.6. Let P be a generic quadratic portrait such that X1(P) has genus 1 or 2. Fix +a prime p ∈ Spec Z, and consider the set +Sp,P := {β ∈ X1(P)(Q, 2) : vp(c(β)) ≥ 0}. +For all but finitely many β ∈ Sp,P, we have G(fc(β), Q(β)) ∼= P. +Proof. Suppose β ∈ Sp,P, set c := c(β), and let K := Q(β). Removing at most finitely many +points β ∈ Sp,P, we may assume that G(fc, K) is generic quadratic. Since β ∈ X1(P)(K) +and G(fc, K) is generic quadratic, G(fc, K) has a subportrait isomorphic to P. +Let p ∈ Spec OK be a prime lying above p. Since p has norm at most p2, the map fc +has no K-rational points of period larger than B := B(p2). Thus, as explained following +Proposition 5.1, if G(fc, K) ̸∼= P, then G(fc, K) must contain one of finitely many portraits +P1, . . . , Pn properly containing P. But each of the curves X1(Pi) has only finitely many +quadratic points by Corollary 5.5; this completes the proof. +□ +Proposition 5.7. Let P ∈ Γrat. If K is a quadratic field and c ∈ K satisfies G(fc, K) ∼= P, +then c ∈ Q. +Proof. For P ∈ {8(4), 10(3, 1, 1), 10(3, 2)}, this follows immediately from Theorems 3.16, +3.25, and 3.28 of [13]. For the remaining portraits P—namely, 8(1,1)a and 8(2)a—the proofs +are similar, so we only provide the details for 8(1,1)a. +Let P = 8(1, 1)a. As shown in Appendix A, X1(P) is isomorphic to the elliptic curve E +with affine model y2 = x3 − x2 + x, which is the curve labeled 24A4 in [9] and 24.a5 in [31]. +We claim that if P = (x, y) is a quadratic point on E, then c(P) = − (x2+1)2 +4x(x−1)2 ∈ Q. This is +certainly the case if x ∈ Q, so assume that x is quadratic. +By Lemma 3.3, there exist P0 = (x0, y0) ∈ E(Q) ∖ ∞ and t ∈ Q such that +(5.1) +x2 + (x0 − t2 − 1)x + (x2 +0 + t2x0 − x0 − 2y0t + 1) = 0. +The only affine rational points (x0, y0) on E are (0, 0) and (1, ±1), and for each of these +points we can use (5.1) to rewrite c(P) as a function of t and x which has degree at most 1 +in x. For the points (0, 0), (1, 1), and (1, −1), respectively, we get +c = − +(t2 + 1)2 +4(t − 1)(t + 1), +c = −t2 (t2 − 2t + 2) +4(t − 1)2 +, and +c = −t2 (t2 + 2t + 2) +4(t + 1)2 +. +In any case, since t ∈ Q, we must also have c ∈ Q. +□ + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +19 +Proposition 5.8. For all P ∈ Γ0 ∪ Γquad there exist infinitely many c ∈ Q(2) such that +G(fc, Q(c)) ∼= P. +Proof. For the portraits in Γ0, this follows from Proposition 5.4. The proofs for 8(1,1)b and +8(2)b (resp., 10(2,1,1)a and 10(2,1,1)b) are very similar, so we only provide the details for +8(1,1)b, 10(2,1,1)a, and 8(3). +First, let P be the portrait 8(1,1)b. Appendix A shows that X1(P) is isomorphic to the +curve X with affine model y2 = 2(x3 + x2 − x + 1), with c : X → P1 given by +c = − +2(x2 + 1) +(x + 1)2(x − 1)2. +Set P0 = (x0, y0) := (1, 2) ∈ X(Q). For t ∈ Q, the line y − 2 = t(x − 1) intersects the curve +X at P0 and two additional points Pt and P t. Since t and P0 are rational, either Pt and P t +are rational as well, or Pt and P t are quadratic conjugates. Since X(Q) is finite, we may, +at the expense of excluding finitely many t, assume that Pt and P t are quadratic Galois +conjugates. Let K := Q(Pt), and let τ be the nontrivial element of Gal(K/Q). With this +setup, we have Pt + P t = −P0 ̸= O, so y(Pt) ̸= −y(P t) = −y(Pt)τ, and therefore x(Pt) /∈ Q. +By calculating the intersection of y − 2 = t(x − 1) with the affine curve X, we find that the +minimal polynomial of x(Pt) must be +x2 − t2 − 4 +2 +x + t2 − 4t + 2 +2 +. +Thus, we may rewrite c(Pt) as +c(Pt) = +t + 2 +8(t − 2)x(Pt) − t5 − 10t3 + 8t2 + 8t + 32 +16(t − 2)2t +. +Finally, for any t ∈ Q with t ≡ 1 (mod 3), we see that x(Pt) and c(Pt) are integral at p = 3, +so Pt ∈ S3,P. By Lemma 5.6, we conclude that there are infinitely many quadratic c with +G(fc, Q(c)) ∼= P. +Next, we let P be the portrait 10(2,1,1)a, and we take X to be the curve defined by +y2 + xy + y = x3 − x2 − x with map c : X → P1 given by +c = +x − 2 +4x(x − 1)y − x4 − x3 + 3x − 1 +4x2(x − 1) +. +By Hilbert irreducibility, there are infinitely many points on X with x ∈ Q, x ≡ 2 (mod 3), +and y /∈ Q. For all such points P = (x, y), c(P) must be quadratic, and we have P ∈ S3,P. +The desired conclusion again follows from Lemma 5.6. +Finally, we let P be the portrait 8(3). Let X be the curve y2 = x6−2x4+2x3+5x2+2x+1 +with c : X → P1 given by +c = −x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1 +4x2(x + 1)2 +. +The curve X has two rational points at infinity; we denote these by ∞+ and ∞−. These two +points are transposed by the hyperelliptic involution ι : X → X given by ι(x, y) = (x, −y). +Let J be the Jacobian of the genus-2 curve X. For a thorough treatment of the arithmetic +of genus-2 curves, we recommend [7]; here, we just summarize the necessary properties. + +20 +JOHN R. DOYLE AND DAVID KRUMM +Points on J correspond to degree-0 divisor classes on X. Moreover, by the Riemann–Roch +theorem, every nontrivial divisor class can be written uniquely as +{P, Q} := [P + Q − ∞+ − ∞−] +with P, Q ∈ X and Q ̸= ι(P), up to swapping P and Q. (The trivial class O is equal to +{P, ι(P)} for all P ∈ X.) A point {P, Q} ̸= O is rational if and only if either P and Q are +both rational themselves, or P and Q are Galois-conjugate quadratic points. +Fix the point +P0 := +� +−1 +4 +� +1 + +√ +−15 +� +, − 1 +16 +� +17 + 9 +√ +−15 +�� +∈ X(Q, 2), +for which we have c(P0) = +1 +48 +� +7 + 8√−15 +� +/∈ Q. If we let +P 0 := +� +−1 +4 +� +1 − +√ +−15 +� +, − 1 +16 +� +17 − 9 +√ +−15 +�� +be the Galois conjugate of P0, then D0 := {P0, P 0} ∈ J(Q). Note that P 0 ̸= ι(P0), so +D0 ̸= O. Poonen showed in [43, Prop. 1] that J(Q) ∼= Z, so D0 has infinite order. +The curve X (hence also its Jacobian J) has good reduction at the prime p = 7. A straight- +forward computation (e.g., in Magma) shows that the reduction �D0 ∈ J(F7) has order 21, +so Dn := (1 + 21n)D0 ≡ D0 (mod 7) for all n ∈ Z. For each n we write Dn = {Pn, P n} +with Pn ≡ P0 (mod 7) and P n ≡ P 0 (mod 7). We claim that Pn ∈ S7,P for all n ∈ Z, from +which the result follows by Lemma 5.6. +Since −15 ≡ −1 (mod 7) is not a square in F7, � +P0 is quadratic over F7, thus the same is +true for � +Pn for all n ∈ Z. This implies that Pn is quadratic over Q for all n. +The map c : X → P1 has good reduction at p = 7, so Pn ≡ P0 (mod 7) implies that +c(Pn) ≡ c(P0) (mod 7). Arguing as in the previous paragraph, we conclude that c(Pn) +is quadratic over Q. Finally, we note that since c(Pn) ≡ c(P0) ̸≡ ∞ (mod 7), we have +v7(c(Pn)) ≥ 0, so Pn ∈ S7,P. +□ +5.3. Proofs of Theorems 1.6 and 1.7. +Proof of Theorem 1.6. That (ii) implies (i) is precisely the second statement in Proposi- +tion 5.2, so it remains only to show that (i) implies (ii). +Assume there are infinitely many c ∈ Q such that G(fc, Q) ⊊ G(fc, K) ∼= P for some +quadratic field K. +Every such occurrence of P as G(fc, K) yields a quadratic point on +Y1(P), so there are infinitely many quadratic points on X1(P). Thus P ∈ Γ by Theorem 4.1. +All that remains for us to show is that P cannot be isomorphic to ∅ or 6(3). Certainly +one cannot have G(fc, Q) ⊊ G(fc, K) ∼= ∅, so we need only show that if c ∈ Q and K is a +quadratic field with G(fc, K) ∼= 6(3), then in fact G(fc, Q) ∼= 6(3). +Supposing that G(fc, K) ∼= 6(3), the map fc then has a period-3 point α ∈ K, and the six +K-rational preperiodic points prescribed by the portrait 6(3) are ±α, ±fc(α), and ±f 2 +c (α). +It therefore suffices to show that α ∈ Q. +Let τ be the nontrivial element of the Galois group Gal(K/Q). Since fc is defined over Q, +ατ is also a K-rational point of period 3, hence lies in the cycle {α, fc(α), f 2 +c (α)} (because + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +21 +the portrait 6(3) has only one 3-cycle2). Write ατ = f k +c (α) for some k ∈ {0, 1, 2}. Then +α = (ατ)τ = f k +c (ατ) = f 2k +c (α). +Since α has exact period 3, this implies that k = 0; that is, ατ = α. Thus α ∈ Q, and +therefore G(fc, Q) ∼= 6(3). +□ +Proof of Theorem 1.7. First suppose that (i) holds; i.e., there are infinitely many c ∈ Q(2) ∖ Q +such that G(fc, Q(c)) ∼= P. The curve X1(P) then has infinitely many quadratic points, and +therefore P ∈ Γ by Theorem 4.1. By Proposition 5.7, we must have +P ∈ Γ ∖ Γrat = Γ0 ∪ Γquad. +Thus, (i) implies (ii). The converse is precisely Proposition 5.8. +□ +6. Fields of definition of quadratic points +In this section we address the question of whether the existence of a given preperiodic +portrait over a given quadratic field has implications regarding standard arithmetic invariants +of that field. In particular, we focus on the portraits that occur infinitely often over quadratic +fields, namely those in the set Γ. +To be precise, we are interested in arithmetic invariants of the fields of definition of qua- +dratic points on dynamical modular curves X1(P). To partially justify the transition from +realizations of portraits to simply points on dynamical modular curves, we begin by proving +the following result: +Proposition 6.1. For a number field K and a generic quadratic portrait P, the following +are equivalent: +(i) There exist infinitely many c ∈ K such that G(fc, K) ∼= P. +(ii) The curve X1(P) has infinitely many K-rational points. +Proof. That (i) implies (ii) is immediate from the definition of X1(P), so suppose that +X1(P)(K) is infinite. Since X1(P) is irreducible (see Proposition 2.6(a)), this implies that +X1(P) is isomorphic over K either to P1 or to an elliptic curve. +In the former case, the result follows from essentially the same proof as for Proposition 5.4. +In fact, the appropriate modification of that proof shows that there are infinitely many c ∈ K +such that Q(c) = K and such that G(fc, K) ∼= P. +In the latter case, choose a non-torsion point Q0 ∈ X1(P)(K), and choose a prime p ∈ +Spec OK of good reduction for the morphism c : X1(P) → P1 such that vp(Q0) ≥ 0. Since Q0 +is non-torsion, there are infinitely many points Q ∈ X1(P)(K) such that Q ≡ Q0 (mod p), +and since the morphism c has good reduction at p, we have c(Q) ≡ c(Q0) (mod p) for every +such Q. In particular, we have infinitely many Q ∈ X1(P)(K) such that vp(c(Q)) ≥ 0, so +the desired result follows from Lemma 5.6. +□ +We now move on to a result concerning the splitting of rational primes in the quadratic +fields over which the portrait 10(3, 1, 1) is realized as a preperiodic portrait. This example +is included in order to illustrate our methods, but similar reasoning can be applied to any +portrait in Γ for which the corresponding modular curve is hyperelliptic. +2In fact, if K is any quadratic field and c ∈ K, then fc has at most one 3-cycle defined pointwise over K. +This follows from [36, Thm. 3] when c ∈ Q and [10, Thm. 4.5] in general. + +22 +JOHN R. DOYLE AND DAVID KRUMM +As noted in [43], the dynamical modular curve corresponding to the portrait 10(3, 1, 1) +is isomorphic to Xell +1 (18). If K is the field of definition of a quadratic point on this curve, +Kenku and Momose [26, Prop. 2.4] show that +• either 2 splits or 3 does not split in K; +• 3 is not inert in K; and +• 5 and 7 are unramified in K. +In what follows, for every polynomial f ∈ Z[x], we denote by πf the set of all integer primes +p such that f does not have a root modulo p. Extending the results of Kenku and Momose, +we prove the following. (Note that this proves Theorem 1.10.) +Theorem 6.2. Let K be a quadratic field such that G(fc, K) ∼= 10(3, 1, 1) for some c ∈ K. +Then the prime 2 splits in K, 3 is not inert in K, and letting +f(x) = x6 + 2x5 + 5x4 + 10x3 + 10x2 + 4x + 1, +every prime in the set πf (which includes 5 and 7) is unramified in K. Moreover, πf has +Dirichlet density 13/18. +For every nonzero rational number r, we let sqf(r) denote the squarefree part of r, i.e., +the unique squarefree integer d such that r/d is the square of a rational number. +Lemma 6.3. Let f ∈ Z[x] be a monic polynomial of even degree, and let p be an odd prime. +If p ∈ πf, then p is unramified in every quadratic field of the form Q( +� +f(r)) with r ∈ Q. +Proof. Given a quadratic field K = Q( +� +f(r)), we must show that p does not divide the +discriminant of K. Let D = sqf(f(r)), so that K = Q( +√ +D). Since p is odd, it suffices to +show that p does not divide D. Set g(x, y) = y2kf(x/y) ∈ Z[x, y], where deg(f) = 2k, and +write r = n/d with gcd(n, d) = 1. Then D = sqf(g(n, d)), so that +g(n, d) = Ds2, +s ∈ Z. +We now consider two cases. If d ≡ 0 mod p, then the above equation can be reduced +modulo p to obtain n2k ≡ Ds2 mod p. Since p cannot divide n (given that n and d are +coprime), we conclude that p does not divide D, as required. +Suppose now that d ̸≡ 0 mod p. We can then consider the equation d2kf(n/d) = Ds2 as +taking place in the ring Zp. If p | D, then reducing modulo p we obtain f(n/d) ≡ 0 mod p, +contradicting the hypothesis that f has no root modulo p. Therefore p cannot divide D. +□ +Proof of Theorem 6.2. By [13, Thm. 3.25], we have K = Q( +� +f(r)) for some r ∈ Q∖{0, −1}. +Writing r = n/d in lowest terms, it follows that K = Q( +� +g(n, d)), where +g(n, d) := d6f(n/d) = n6 + 2n5d + 5n4d2 + 10n3d3 + 10n2d4 + 4nd5 + d6. +We claim that g(n, d) ≡ 1 mod 8. If n, d are both odd, then +g(n, d) ≡ 1 + 2nd + 5 + 10nd + 10 + 4nd + 1 = 17 + 16nd ≡ 1 mod 8. +If n is even and d is odd, then g(n, d) ≡ d6 ≡ 1 mod 8. If n is odd and d is even, then +g(n, d) ≡ 1 + 2nd + 5d2 mod 8. Writing n = 2k + 1 for some integer k we see that +g(n, d) ≡ 5d2 + 2d + 1 ≡ (d + 1)2 ≡ 1 mod 8, +which proves the claim. Letting D = sqf(g(n, d)), the fact that g(n, d) ≡ 1 (mod 8) implies +that D ≡ 1 mod 8 and therefore 2 splits in Q( +√ +D) = K. + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +23 +Similar reasoning shows that g(n, d) is congruent to either 0 or 1 modulo 3. Considering +all possible values of n and d modulo 9, we find that if g(n, d) is divisible by 9, then n and +d are both divisible by 3, which is a contradiction; hence 9 ∤ g(n, d). Writing g(n, d) = Ds2 +for some integer s, this implies that s is not divisible by 3, and therefore g(n, d) ≡ D mod 3. +Hence, D is congruent to 0 or 1 modulo 3, and therefore 3 is not inert in K. +A computation in Magma based on [27, Thm. 2.1] shows that πf has Dirichlet density +13/18. Finally, Lemma 6.3 implies that every odd prime in πf is unramified in K, and we +have already shown that 2 is unramified in K. +□ +An argument very similar to the proof of Theorem 6.2 yields the following result, in which +the relevant dynamical modular curve is known to be isomorphic to Xell +1 (13). +Theorem 6.4. Let K be a quadratic field such that G(fc, K) ∼= 10(3, 2) for some c ∈ K. +Then the prime 2 splits in K, and every prime in πf is unramified in K, where +f(x) = x6 + 2x5 + x4 + 2x3 + 6x2 + 4x + 1. +Moreover, the set πf has Dirichlet density 13/18. +Our next result concerns the curve Xell +1 (16), which is isomorphic to the modular curve for +the portrait 8(4); see Section 3.7 of [13]. In contrast to Theorems 6.2 and 6.4, we show that +the discriminants of quadratic fields defined by points on Xell +1 (16) are not restricted to any +residue class. (Note that this proves Theorem 1.9(a).) +Theorem 6.5. Let P denote the portrait 8(4). For every prime integer p and every residue +class c ∈ Z/pZ, there exist infinitely many squarefree integers d ∈ c such that the curve +X1(P) has a quadratic point defined over the field Q( +√ +d). +For the proof of the theorem we use the methods of [28] and [29]; the following lemma, +which follows from Proposition 14 in [29], collects the main tools to be used. +Lemma 6.6. Let f ∈ Z[x] be a squarefree polynomial of degree at least 3, and such that +every irreducible factor of f has degree at most 6. Let +S(f) = {sqf(f(x)) : x ∈ Q and f(x) ̸= 0}. +Let D be the largest integer dividing all integer values of f. Fix a prime p such that f has +an irreducible factor whose discriminant is not divisible by p, and let ε = εp ∈ {0, 1} be +the parity of ordp(D). Finally, for c ∈ Z/pZ and v ∈ Z, let σ(p, c, v) denote the following +statement. +(6.1) +σ(p, c, v) : +� +� +� +� +� +There exist h ∈ c and x0, y0 ∈ Z satisfying +• hy2 +0 ≡ f(x0) (mod p2(v+ε)+1) and +• ordp(y0) = v + ε. +Suppose c is nonzero and σ(p, c, v) holds for some v ≥ 0. Then the set S(f) ∩ c is infinite. +Proof of Theorem 6.5. By [38, p. 93], the curve X1(P) is hyperelliptic and has an affine +model given by y2 = f(x), where f(x) = −x(x2 + 1)(x2 − 2x − 1). +In order to prove the theorem it suffices to show that, for every prime p and residue class +c ∈ Z/pZ, the set S(f)∩c is infinite. Indeed, if d belongs to this set, we may write dy2 +0 = f(x0) +for some rational numbers x0, y0 with y0 ̸= 0. The pair (x0, y0 +√ +d) then represents a quadratic +point on X1(P) whose field of definition is Q( +√ +d). Hence, the theorem follows. + +24 +JOHN R. DOYLE AND DAVID KRUMM +In the notation of Lemma 6.6, for the above polynomial f(x) we have D = 2; moreover, +the irreducible factor x of f(x) has discriminant 1, which is coprime to every prime p. It +follows in particular that +εp = +� +0 +if p is odd, +1 +if p = 2. +Fix a prime p. For the class c = 0 ∈ Z/pZ, Theorem 2.1 in [28] implies that the set +S(f) ∩ c is infinite, as desired. (When p = 2, the hypotheses of the cited theorem are not all +satisfied, but the proof still applies.) Next, we claim that +for every nonzero c ∈ Z/pZ, either σ(p, c, 0) or σ(p, c, 1) must hold. +Assuming this claim for the moment, Lemma 6.6 implies that the set S(f) ∩ c is infinite, +completing the proof of the theorem. +To prove the claim we consider first the case p = 2: taking c = 1, the statement σ(p, c, 1) +can be shown to hold by setting (h, x0, y0) = (1, 16, 4) in (6.1). The remainder of the proof +is divided into three cases. +Case p ≤ 5: Taking p = 3, we check that σ(p, c, 1) holds for c = 1, 2 by using the tuples +(h, x0, y0) = (1, 9, 3) +and +(h, x0, y0) = (2, 18, 3). +Similarly, taking p = 5, we check that σ(p, c, 1) holds for c = 1, 2, 3, 4, respectively, by using +the following tuples (h, x0, y0): +(1, 25, 5), (2, 18, 5), (3, 75, 5), (4, 7, 5). +In the remaining two cases we show that σ(p, c, 0) holds. For r ∈ c, let Xr be the hy- +perelliptic curve over Fp defined by the equation ry2 = f(x). Since εp = 0, the statement +σ(p, c, 0) is equivalent to the requirement that Xr have an affine point (x0, y0) ∈ Xr(Fp) with +y0 ̸= 0; we refer to such points as nontrivial points on Xr. Thus, it remains to show that Xr +has at least one nontrivial point. +Case 7 ≤ p ≤ 23: A straightforward search for points verifies that #Xr(Fp) ≥ 7 for every +nonzero r ∈ Fp. (Note that it suffices to check this for just two values of r, one in each +square class modulo p.) The number of affine points (x0, y0) ∈ Xr(Fp) having y0 = 0 is at +most 5, so there must exist at least one nontrivial point in Xr(Fp), as required. +Case p ≥ 29: For r ∈ Fp ∖ 0, the curve Xr has genus 2, so the Hasse–Weil bound yields +#Xr(Fp) ≥ ⌊p + 1 − 4√p⌋ ≥ 7. +The same reasoning as in the previous case implies that Xr has a nontrivial Fp-point. +□ +We end the paper by proving Theorem 1.9(b). +Proposition 6.7. Let P = 8(4). There exist infinitely many imaginary quadratic fields K +with class number divisible by 10, such that X1(P) has a quadratic point defined over K. +Proof. As noted earlier, the curve X1(P) is isomorphic to Xell +1 (16). The result follows from +[20, Cor. 3.2], since the Jacobian J1(16) has a rational torsion point of order 10. +□ +Remark 6.8. Experimental evidence supports a statement stronger than Proposition 6.7: +for every imaginary quadratic field K ̸= Q(√−15) that is the field of definition of a point on +X1(P), the class number of K is divisible by 10. One approach to proving this is suggested +by the methods of [2, 20]; however, the required computational tools (in particular, for +computing quotients of abelian varieties) do not seem to be presently available. + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +25 +Appendix A. Dynamical modular curves of genera 1 and 2 +We provide here models for all dynamical modular curves X1(P) of genus 1 or 2, together +with an explicit description of the morphism c : X1(P) → P1. Each of these models appears +in [43]. Note that in some cases we provide two models—one of the form y2 = F(x), and +another that turns out to be more useful for certain aspects of our proofs. +Portrait P +Model(s) for X1(P) +Morphism c : X1(P) → P1 +8(1,1)a +y2 = x3 − x2 + x +− (x2 + 1)2 +4x(x − 1)2 +8(1,1)b +y2 = 2(x3 + x2 − x + 1) +− +2(x2 + 1) +(x + 1)2(x − 1)2 +8(2)a +y2 = x3 − 2x + 1 +−(x2 − 2x + 2)(x2 + 2x − 2) +4x2(x − 1) +8(2)b +y2 = 2(x3 + x2 − x + 1) +−x4 + 2x3 + 2x2 − 2x + 1 +(x + 1)2(x − 1)2 +10(2,1,1)a +y2 = 5x4 − 8x3 + 6x2 + 8x + 5 +−(3x2 + 1)(x2 + 3) +4(x + 1)2(x − 1)2 +y2 + xy + y = x3 − x2 − x +x − 2 +4x(x − 1)y − x4 − x3 + 3x − 1 +4x2(x − 1) +10(2,1,1)b +y2 = (5x2 − 1)(x2 + 3) +−(3x2 + 1)(x2 + 3) +4(x + 1)2(x − 1)2 +y2 + xy + y = x3 + x2 +− +x + 2 +4x(x + 1)y − x4 + 4x3 + 6x2 + 3x + 1 +4x2(x + 1) +8(3) +y2 = x6 − 2x4 + 2x3 + 5x2 + 2x + 1 +−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1 +4x2(x + 1)2 +8(4) +y2 = −x(x2 + 1)(x2 − 2x − 1) +(x2 − 4x − 1)(x4 + x3 + 2x2 − x + 1) +4x(x + 1)2(x − 1)2 +10(3,1,1) +y2 = x6 + 2x5 + 5x4 + 10x3 + 10x2 + 4x + 1 +−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1 +4x2(x + 1)2 +10(3,2) +y2 = x6 + 2x5 + x4 + 2x3 + 6x2 + 4x + 1 +−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1 +4x2(x + 1)2 + +26 +JOHN R. DOYLE AND DAVID KRUMM +Appendix B. Tables of preperiodic portraits +B.1. Preperiodic portraits realized over quadratic fields. We list here the 46 portraits +known to be realized as G(fc, K) for some quadratic field K and c ∈ K. These were found in +the search described in [13]. The label of each portrait is in the form N(ℓ1, ℓ2, . . .), where N +is the number of vertices in the portrait and ℓ1, ℓ2, . . . are the lengths of the directed cycles +in the portrait in nonincreasing order. If more than one isomorphism class with this data +was observed, we add a lowercase Roman letter to distinguish them. For example, the labels +5(1,1)a and 5(1,1)b correspond to the two isomorphism classes of portraits observed that +have five vertices and two fixed points. In all figures, we omit the vertex corresponding to +the fixed point at infinity. +0 +2(1) +3(1,1) +3(2) +4(1) +4(1,1) +4(2) +5(1,1)a +5(1,1)b +5(2)a +5(2)b +6(1,1) +6(2) +6(2,1) +6(3) +7(1,1)a +7(1,1)b +7(2,1,1)a + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +27 +7(2,1,1)b +8(1,1)a +8(1,1)b +8(2)a +8(2)b +8(2,1,1) +8(3) +8(4) +9(2,1,1) +10(1,1)a +10(1,1)b +10(2) +10(2,1,1)a +10(2,1,1)b +10(3)a +10(3)b +10(3,1,1) + +28 +JOHN R. DOYLE AND DAVID KRUMM +10(3,2) +12(2) +12(2,1,1)a +12(2,1,1)b +12(3) +12(4) +12(4,2) +12(6) +14(2,1,1) +14(3,1,1) +14(3,2) + +QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES +29 +B.2. Additional portraits. The following portraits are not known to be realized over qua- +dratic fields (and, in some cases, have been shown not to be); however, they make an +appearance in the discussion in Section 4.2, so we include them here. The labels G1, . . . , G10 +are taken from [10]. +G1 +G2 +G3 +G4 +G5 +G6 +G7 +G8 +G9 +G10 + +30 +JOHN R. 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Pezda, Polynomial cycles in certain local domains, Acta Arith. 66 (1994), no. 1, 11–22. MR 1262650 +[43] Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over Q: a refined +conjecture, Math. Z. 228 (1998), no. 1, 11–29. MR 1617987 (99j:11076) +[44] Jean-Pierre Serre, Topics in Galois theory, second ed., Research Notes in Mathematics, vol. 1, A K +Peters, Ltd., Wellesley, MA, 2008, With notes by Henri Darmon. MR 2363329 +[45] Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, +Springer, New York, 2007. MR 2316407 (2008c:11002) +[46] Henning Stichtenoth, Algebraic function fields and codes, second ed., Graduate Texts in Mathematics, +vol. 254, Springer-Verlag, Berlin, 2009. MR 2464941 (2010d:14034) +[47] Michael Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 +(2008), 367–380. 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MR 1270956 (95a:11025) +[49] Michael Zieve, Cycles of polynomial mappings, Ph.D. thesis, University of California, Berkeley, 1996. +Department of Mathematics, Oklahoma State University, Stillwater, OK 74078 +Email address: john.r.doyle@okstate.edu +URL: http://maths.dk +Email address: david.krumm@gmail.com + diff --git a/19AyT4oBgHgl3EQfofjr/content/tmp_files/load_file.txt b/19AyT4oBgHgl3EQfofjr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fd6bc665547bd1bc6bf9c7988d30c65b53b842a --- /dev/null +++ b/19AyT4oBgHgl3EQfofjr/content/tmp_files/load_file.txt @@ -0,0 +1,1273 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf,len=1272 +page_content='QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Introduction Let K be a field with algebraic closure K, and let f be a rational function in one variable over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Corresponding to f there are a morphism of algebraic varieties P1 K → P1 K and a map on point sets P1(K) → P1(K), both of which we also denote by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A point P ∈ P1(K) is called periodic for f if there exists a positive integer n such that f n(P) = P, where f n denotes the n-fold composition of f with itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' in that case, the smallest such n is called the (exact) period of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' More generally, the point P is preperiodic for f if there exists m ≥ 0 such that f m(P) is periodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' the smallest such m is then called the preperiod of P, and we call the period of f m(P) the eventual period of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Here, f 0 is interpreted as the identity map, so that periodic points are considered preperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For any intermediate field K ⊆ L ⊆ K we define a directed graph G(f, L), called the preperiodic portrait of f over L, whose vertices are the points P ∈ P1(L) that are preperiodic for f, and whose directed edges are the ordered pairs (P, f(P)) for all vertices P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the terminology of graph theory, G(f, L) is a functional graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', a directed graph in which every vertex has out-degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Throughout this paper we will use the term portrait instead of functional graph in order to emphasize our dynamical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Portraits for quadratic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Assume henceforth that K is a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We will primarily, though not exclusively, be interested in the case where K is a quadratic extension of Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' we refer to such fields simply as quadratic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A type of problem that has received much attention in the field of arithmetic dynamics is that of classifying the portraits G(f, K) up to graph isomorphism as f is allowed to vary in an infinite family of rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' An early example of this classification problem is Poonen’s study [43] of the portraits G(f, Q) as f varies over the family of all quadratic polynomials with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1 (Poonen [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Assume that there is no quadratic polynomial over Q having a rational periodic point of period greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then, for every quadratic polynomial f ∈ Q[z], the portrait G(f, Q) is isomorphic to one of the following twelve graphs (using the labels from Appendix B): ∅, 2(1), 3(1, 1), 3(2), 4(1, 1), 4(2), 5(1, 1)a, 6(1, 1), 6(2), 6(3), 8(2, 1, 1), 8(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Primary 37P05, 37P35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Secondary 37P15, 11G30, 14G05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Arithmetic dynamics, dynatomic curve, preperiodic portrait, uniform bounded- ness conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='00510v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='NT] 2 Jan 2023 2 JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM Regarding the assumption in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1, it is known that a quadratic polynomial over Q cannot have rational periodic points of period 4 (Morton [38]), period 5 (Flynn–Poonen– Schaefer [18]), or, assuming that the conclusions of the Birch and Swinnerton-Dyer conjecture hold for a certain Jacobian variety, period 6 (Stoll [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In addition, a substantial amount of empirical evidence supporting the assumption in Poonen’s theorem has been provided by Hutz and Ingram [23] and Benedetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' However, it remains an open problem to prove that this assumption is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Portraits for other families of quadratic maps over Q are studied in the articles [3,6,32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The present paper concerns the preperiodic portraits of quadratic polynomials defined over quadratic fields, a topic previously explored in [10,12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Analogy with torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A guiding principle that has proved fruitful in arith- metic dynamics is to regard the set of preperiodic points of a map as being analogous to the set of torsion points on an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, for instance, the well-known fact that the set of K-rational torsion points on an abelian variety is finite is viewed as analogous to a theorem of Northcott [41] stating that, for every rational function f over K of degree at least 2, the set of K-rational preperiodic points of f is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Motivated by this analogy, Morton and Silverman formulated the following dynamical analogue of a standard uniform boundedness conjecture for abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We state the dynamical conjecture only in the case of endomorphisms of the projective line, although a similar statement applies to arbitrary projective spaces1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2 (Morton–Silverman [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a number field K and morphism f : P1 K → P1 K of degree greater than 1, the number of K-rational preperiodic points of f is bounded above by a constant depending only on the degree of f and the absolute degree of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This dynamical uniform boundedness conjecture would imply, in particular, that there are only finitely many isomorphism classes of portraits G(f, Q) as f ranges over all quadratic polynomials with rational coefficients, since the number of vertices in such a portrait is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1 can thus be seen as a refinement of the conjecture in this case, as it provides a (conditionally) complete list of all possible portraits for the family of quadratic polynomial maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the analogy with torsion points, Poonen’s list of portraits corresponds to the list of abelian groups that can be realized as the torsion subgroup of an elliptic curve over Q, the latter list being provided by a well-known theorem of Mazur [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Similarly, the Morton–Silverman conjecture would imply that the portraits G(f, K), where K is a quadratic field and f is a quadratic polynomial over K, fall into finitely many iso- morphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A conjecturally complete list of classes was first proposed in [13], and is included here in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The list is comprised of 46 portraits, and can be viewed as anal- ogous to the list of 26 abelian groups, known by work of Kamienny [25] and Kenku–Momose [26], that can arise as torsion subgroups of elliptic curves over quadratic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Infinitely occurring portraits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Our primary objective in this paper is to determine, under a suitable notion of equivalence of maps, which of the 46 graphs in [13] arise as the preperiodic portrait of infinitely many inequivalent quadratic polynomials over quadratic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the context of elliptic curves, both over Q and over quadratic fields, the correspond- ing question is well understood: every abelian group that arises as the torsion subgroup of an elliptic curve can be realized as such by infinitely many non-isomorphic curves (see [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1Interestingly, work of Fakhruddin [17] shows that the more general Morton–Silverman conjecture in fact implies its analogue for abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' See also [45, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3], where Merel’s theorem for elliptic curves is shown to follow from Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2 QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES 3 In order to state our questions more precisely, we begin by defining the appropriate equiv- alence relation on maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Two morphisms f, h : P1 K → P1 K are called linearly conjugate over K if there exists an automorphism σ ∈ PGL2(K) such that h = σ−1 ◦ f ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (Similarly, one can define linear conjugacy over any extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=') In that case, a simple argument shows that the portraits G(f, K) and G(h, K) are isomorphic as directed graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' hence, the isomorphism class of G(f, K) is determined by the linear conjugacy class of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the case of quadratic polynomials, it is well known that every such map f ∈ K[z] is linearly conjugate to a unique map of the form fc(z) := z2 + c, where c ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, in studying the portraits of quadratic polynomials we may restrict attention to the one-parameter family of maps fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Returning to the question of portraits arising infinitely often, the case of quadratic poly- nomials over Q was answered by Faber, who showed in addition that Poonen’s list in [43] does not omit any such portrait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3 (Faber [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a portrait P, the following are equivalent: (i) There exist infinitely many c ∈ Q such that G(fc, Q) ∼= P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (ii) P is isomorphic to one of the following graphs (using the labels from Appendix B): ∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Motivated by Faber’s theorem, we now state the main questions to be addressed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Among the 46 known isomorphism classes of portraits arising as G(fc, K), with K a quadratic field and c ∈ K, which ones can be realized as such by infinitely many algebraic numbers c?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In addition, must every infinitely occurring portrait belong to one of the 46 known isomorphism classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We define the following sets of portraits using labels as in Appendix B: Γ0 := {∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Γrat := {8(1, 1)a, 8(2)a, 8(4), 10(3, 1, 1), 10(3, 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Γquad := {8(1, 1)b, 8(2)b, 8(3), 10(2, 1, 1)a/b};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Γ := Γ0 ∪ Γrat ∪ Γquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We provide two answers to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4 which differ in their level of specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The simplest is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5, with Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7 providing additional information in terms of the above subsets of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For an integer n ≥ 1, we define Q(n) := {α ∈ Q : [Q(α) : Q] ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a portrait P, the following are equivalent: (i) There exist infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field K containing c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (ii) P ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 4 JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5 can be refined in order to take into account certain subtleties illustrated by the following example: We see from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3 that the portrait P = 4(2) is realized as G(fc, Q) for infinitely many c ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For each such c, the set of preperiodic points for fc is a set of bounded height, and therefore fc has only finitely many preperiodic points of algebraic degree 2 over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Hence, for each of the infinitely many c ∈ Q with G(fc, Q) ∼= P, we must also have G(fc, K) ∼= P for all but finitely many quadratic fields K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We therefore show that each of the portraits P ∈ Γ is realized infinitely often—even if one excludes the infinitely many “trivial” realizations in the sense of the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This is done in the next two theorems, which are stated separately in order to distinguish between polynomials with rational coefficients and those with quadratic algebraic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a portrait P, the following are equivalent: (i) There exist infinitely many c ∈ Q such that G(fc, Q) ⊊ G(fc, K) ∼= P for some quadratic field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (ii) P ∈ Γ ∖ {∅, 6(3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a portrait P, the following are equivalent: (i) There exist infinitely many c ∈ Q(2) ∖ Q such that G(fc, Q(c)) ∼= P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (ii) P ∈ Γ0 ∪ Γquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Note that every portrait in Γ is covered by at least one of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7, since ∅ and 6(3) are elements of Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In particular, these two theorems together imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Quadratic points on dynamical modular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The proofs of our main results rely heavily on the concept of a dynamical modular curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' To each of the 46 portraits from [13], and more generally to any portrait that could potentially be realized as the preperi- odic portrait of a quadratic polynomial over a number field, we associate an algebraic curve parametrizing instances of the portrait as a preperiodic portrait G(fc, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The curve cor- responding to a portrait P will be denoted X1(P) by analogy with the classical modular curves X1(N) parametrizing elliptic curves with a torsion point of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' To avoid confu- sion, the latter curve will henceforth be denoted Xell 1 (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The details of the construction as well as basic properties of dynamical modular curves are discussed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A more general construction of dynamical moduli spaces appears in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The core of our analysis in this paper is a study of basic geometric invariants, such as genus and gonality, of the curves X1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We then turn this geometric data into arithmetic data using Faltings’ theorem on rational points on subvarieties of abelian varieties, via the following result of Harris and Silverman: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='8 (Harris–Silverman [21, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let X be a smooth, irreducible, projective curve of genus g ≥ 2 defined over a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If X is neither hyperelliptic or bielliptic over K, then X has only finitely many points that are quadratic over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In addition, we consider arithmetic questions regarding the fields of definition of quadratic points on X1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In particular, if X1(P) has a point defined over a quadratic number field K, what can be said about basic arithmetic invariants of K, such as discriminant and class number?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For the curves Xell 1 (N), arithmetic questions of this kind have been discussed by several authors: Momose [35] shows that if K is the field of definition of a quadratic point on Xell 1 (13), then the prime 2 splits in K, and 3 is unramified in K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Bosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' [5] show that K must be a real quadratic field, an observation also made in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the case of the QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES 5 modular curves Xell 0 (N), Najman and Trbovi´c [40] prove arithmetic results of this type for several values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For the dynamical modular curves X1(P) we prove the following two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Though our methods can be applied to several portraits in the set Γ (namely, those for which the corresponding modular curve is hyperelliptic), the portraits 8(4) and 10(3,1,1) are highlighted here due to their significance in the context of elliptic curves, explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' By a quadratic point on an algebraic curve over a field k, we mean a point whose field of definition is a quadratic extension of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let P denote the portrait 8(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (a) For every prime p and every residue class c ∈ Z/pZ, there exist infinitely many squarefree integers d ∈ c such that X1(P) has a quadratic point defined over Q( √ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (b) There exist infinitely many imaginary quadratic fields K with class number divisible by 10 such that X1(P) has a quadratic point defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' As noted in [13], the above curve X1(P) is isomorphic to Xell 1 (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='9 provides new information about the collection of quadratic fields K such that there exists an elliptic curve E/K with a K-rational torsion point of order 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Similarly, taking P = 10(3, 1, 1), the curve X1(P) is known to be isomorphic to Xell 1 (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The next theorem strengthens earlier results by Kenku–Momose [26] regarding the splitting of rational primes in the fields of definition of quadratic points on this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let P denote the portrait 10(3, 1, 1) and let K be the field of definition of a quadratic point on X1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (a) The prime 2 splits in K, and 3 is not inert in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (b) There exists an infinite and computable set of primes, denoted π, that is independent of K, and such that every prime in π is unramified in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Points of higher degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Though our primary focus here is on quadratic fields, we make one observation concerning arbitrary number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The next result is a straightforward consequence of a theorem of Frey [19] together with the main theorem of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Fix a positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For any portrait P, let γ(P) denote the set of algebraic numbers c ∈ Q(n) such that P ∼= G(fc, K) for some number field K satisfying c ∈ K ⊂ Q(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' There are only finitely many portraits P such that γ(P) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5 is a more refined version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='11 in the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In Section 2 we define the notion of a generic quadratic portrait and discuss basic facts concerning dynamical modular curves associated to such portraits, followed by general properties of algebraic curves in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In Section 4, we apply geometric arguments to determine all generic quadratic portraits P for which the curve X1(P) has infinitely many quadratic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This proves, in particular, the implication (i) ⇒ (ii) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1 and the immediately preceding discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Section 5 addresses the issue that K-rational points on X1(P) correspond to instances where G(fc, K) simply contains the portrait P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' that is, we need not have an isomorphism G(fc, K) ∼= P, and in fact, in many cases we do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The section culminates with the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7 in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 6 JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM Finally, Section 6 is devoted to arithmetic questions concerning the fields of definition of quadratic points on the curves X1(P), and in particular to proving Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We thank Joe Silverman for helpful comments, and especially for a suggestion that led to the more refined statements in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The first author was partially supported by NSF grant DMS-2112697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Dynamical modular curves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Dynatomic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If f is a polynomial with coefficients in a field K and α ∈ K is a point of exact period n for f, then α is a root of the polynomial f n(z)−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' However, the roots of f n(z) − z may have period strictly dividing n, and indeed there is a factorization f n(z) − z = � d|n Φd,f(z), where (generically) the roots of Φd,f have exact period d for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' M¨obius inversion yields Φn,f(z) = � d|n (f d(z) − z)µ(n/d), where µ denotes the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We call Φn,f the nth dynatomic polynomial of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' More generally, for m, n ≥ 1 we define Φm,n,f(z) := Φn,f(f m(z)) Φn,f(f m−1(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then Φm,n,f is a polynomial whose roots are (again, generically) points of preperiod m and eventual period n for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (That Φn,f and Φm,n,f are indeed polynomials is proven in [22,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=') Since we are specifically interested in the family fc(z) = z2 + c, we write Φn(c, z) := Φn,fc(z) and Φm,n(c, z) := Φm,n,fc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then Φn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', Φm,n) is a polynomial in Z[c, z], and the vanishing locus defines an affine curve Y1(n) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', Y1(m, n)), which we refer to as a dynatomic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, for example, if α has period n for fc, then (c, α) is a point on the dynatomic curve Y1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We denote by X1(·) the normalization of the projective closure of the affine curve Y1(·), and we also refer to X1(·) as a dynatomic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Dynamical modular curves associated to portraits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The dynamical properties of quadratic polynomial maps impose certain restrictions on those portraits that may be realized as G(f, K) for some number field K and a quadratic polynomial f ∈ K[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' First, no point may have more than two preimages under f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Also, for each positive n ∈ Z, the nth dynatomic polynomial for a quadratic polynomial f has degree (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1) D(n) := deg Φn,f(z) = � d|n µ(n/d)2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, a quadratic polynomial has at most D(n) points of period n, partitioned into at most R(n) := D(n)/n cycles of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' With these restrictions in mind, we make the following definition: QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A generic quadratic portrait Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A quadratic portrait is a portrait P satisfying the following properties: (a) Every vertex of P has in-degree at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (b) For each n ≥ 1, the number of n-cycles in P is at most R(n) := 1 n � d|n µ(n/d)2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For any number field K and quadratic polynomial f ∈ K[z], the portrait G(f, K) is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' However, for most quadratic polynomials (in a sense that can be made precise), we can say more about the structure of the set of K-rational preperiodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For the model fc(z) = z2 + c, if α is a preperiodic point for fc, then −α is also preperiodic, since both are preimages of f(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, a preperiodic point typically has either no K-rational preimages or exactly two K-rational preimages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The exception to this rule occurs when α = 0 is a preperiodic point, in which case exactly one preperiodic point (namely, c = fc(0)) has a single K-rational preimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Along the same lines, if a polynomial fc has a K-rational fixed point β, then β is a root of the quadratic polynomial fc(z)−z = z2−z+c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' thus, unless we have disc(fc(z)−z) = 1−4c = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', c = 1/4), there is a second fixed point β′, necessarily defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' With these observations in mind, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A generic quadratic portrait is a quadratic portrait P with the following additional properties: (a) The in-degree of any vertex of P is equal to 0 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (b) If P has a fixed point, then P has exactly two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We will sometimes refer to the results of [11], in which the term “strongly admissible” is used instead of “generic quadratic.” Given a quadratic portrait P, there is a dynamical modular curve Y1(P), defined over Q, whose K-points—for any extension K/Q—correspond to tuples (c, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' , zn) such that z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' , zn are preperiodic points forming a subportrait of G(fc, K) isomorphic to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If Q is the point on Y1(P) corresponding to such a tuple, then the field of definition of Q is Q(c, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We denote by X1(P) the smooth projective curve birational to Y1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' A formal treatment of dynamical modular curves appears in [11], where the curves are de- fined only for generic quadratic portraits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We lose no generality in making such a restriction: Given any quadratic portrait P, there is a unique portrait P′ that is minimal among generic quadratic portraits containing P as a subportrait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In the language of [11], P′ is the generic quadratic portrait generated by the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' It follows from the results of [11, §2] that X1(P) ∼= X1(P′), so we may as well assume that P is generic quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Rather than formally defining X1(P) (we refer the interested reader to [11] or, for a different approach in a more general setting, [15]), we give an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Consider the generic quadratic portrait P appearing in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' One could construct a curve birational to X1(P) simply by giving one equation for each relation coming 8 JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM from an edge in P: if we label the vertices 1, 2, 3, 4 from left to right, we have z2 1 + c = z2, z2 2 + c = z3, z2 3 + c = z2, z2 4 + c = z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2) Note that we must also impose certain Zariski open conditions of the form zi ̸= zj to remove extraneous components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' for example, there is a full component of the curve defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2) on which z1, z2, z3, and z4 are all equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' It is this approach that is taken in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' An alternative approach, which is particular to quadratic polynomials, is to note that any generic quadratic portrait containing a point of period 2 must necessarily contain P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, another (affine) model for X1(P) is the plane curve defined by the vanishing of Φ2(c, z) = (z2 + c)2 + c − z z2 + c − z = z2 + z + c + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In other words, X1(P) is isomorphic to the dynatomic curve X1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This second model has the advantage of being defined in a lower-dimensional affine space (A2, rather than A5), and it is this second approach which is described in detail in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We conclude this example by pointing out that X1(P) ∼= X1(2) also has “degenerate” points where two or more of the vertices of the portrait P collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For example, the equation Φ2(c, z) = 0 has the solution (c, z) = � − 3 4, − 1 2 � despite the fact that − 1 2 is a fixed point for f−3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' However, for a given portrait P, there are only finitely many such degenerate points on X1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Before summarizing the required properties of dynamical modular curves, we recall the following terminology: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let X be a smooth, irreducible projective curve defined over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The k-gonality of X, denoted gonk(X), is the minimal degree of a nonconstant morphism X → P1 defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let P be a generic quadratic portrait, and let k be any field of character- istic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (a) The curve X1(P) is irreducible over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (b) If P′ is a generic quadratic portrait properly contained in P, then there is a finite morphism πP,P′ : X1(P) → X1(P′) of degree at least 2 defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (c) Given any ordering P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' of all generic quadratic portraits, the k-gonalities of the curves X1(Pi) tend to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Parts (a) and (b) are proven in [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='7] and [11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Note that the morphism πP,P′ is obtained simply by forgetting the preperiodic points correspond- ing to vertices of P ∖ P′, hence is defined over the base field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Statement (c) is a slight generalization of, but follows directly from, [14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1(b)], which says that as m + n → ∞, the gonalities of the curves X1(m, n) tend to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Given a bound B, there are only finitely many quadratic portraits P such that every vertex v of P has preperiod m and eventual period n satisfying m + n ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In other words, if for every generic quadratic portrait P we choose a vertex vP with preperiod mP and eventual period nP maximizing the sum mP + nP, we must have mP + nP → ∞ as P ranges over all generic quadratic portraits in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Since there is a nonconstant morphism from X1(P) to X1(mP, nP) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', by part (b)), we have gonk(X1(P)) ≥ gonk(X1(mP, nP)), and the latter expression tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' □ QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES 9 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Fix n ≥ 1 and a portrait P, and suppose there are infinitely many c ∈ Q(n) such that G(fc, K) ∼= P for some degree-n number field K containing c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then the dynamical modular curve X1(P) has infinitely many points of degree at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' It follows from [19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 2] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 5]) that X1(P) must have gonality at most 2n, hence there are only finitely many such portraits P by part (c) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Some useful properties of algebraic curves In this section, we collect a few facts about algebraic curves that will be used throughout the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' First, we provide a statement that follows from Hilbert’s irreducibility theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' see [44, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4] and [30, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let K be a number field, let X be a curve defined over K, and let ϕ : X → P1 be a dominant morphism of degree d ≥ 2 defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then the set T := � P ∈ P1(K) : [K(Q) : K] < d for some Q ∈ ϕ−1(P) � is a thin subset of P1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thin subsets T ⊂ P1(K) have density 0, in the sense that lim N→∞ ���{P ∈ T : h(P) ≤ N} ��� ���{P ∈ P1(K) : h(P) ≤ N} ��� = 0, where h is the na¨ıve Weil height on P1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In particular, for any maximal ideal p ∈ Spec OK and any mod-p residue class c in P1(K), the set c \\ T is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Given an elliptic curve E with Weierstrass equation y2 = f(x), where f ∈ K[x] is square- free of degree 3, it is easy to construct infinitely many quadratic points on E: For “most” x ∈ K, the point (x, y) = (x, � f(x)) is quadratic over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' More precisely, since f is not a square in K[x], it follows from Hilbert irreducibility that f(x0) is a nonsquare in K for all x0 outside a thin subset of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The following result, proven in [13, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2], gives a useful characterization of quadratic points (x, y) with x /∈ Q: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let E/K be an elliptic curve defined by an equation of the form y2 = ax3 + bx2 + cx + d, where a, b, c, d ∈ K and a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Suppose (x, y) ∈ E(K) is a quadratic point with x /∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then there exist (x0, y0) ∈ E(K) and t ∈ k such that y = y0 + t(x − x0) and x2 + ax0 − t2 + b a x + ax2 0 + t2x0 + bx0 − 2y0t + c a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='8, a curve with infinitely many quadratic points must admit a degree-2 morphism to either P1 or an elliptic curve, hence must have gonality at most 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, to prove that a curve has finitely many quadratic points, it suffices to show that the gonality of the curve is greater than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The following inequality is a standard tool for finding lower bounds for gonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 10 JOHN R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' DOYLE AND DAVID KRUMM Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4 (Castelnuovo–Severi inequality [46, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let Y , Y1, and Y2 be curves of genera gY , g1, and g2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Suppose we have maps ϕ1 : Y → Y1 and ϕ2 : Y → Y2 of degrees d1 and d2, and suppose further that there is not an intermediate curve Z and a map ψ : Y → Z of degree greater than 2 such that both ϕ1 and ϕ2 factor through ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1) gY ≤ d1g1 + d2g2 + (d1 − 1)(d2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Dynamical modular curves with infinitely many quadratic points The purpose of this section is to prove one direction of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5, namely that if there are infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field K containing c, then P ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Since any such realization of P as G(fc, K) yields a quadratic point on the dynamical modular curve Y1(P), it suffices to prove the following: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let P be a generic quadratic portrait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then X1(P) has infinitely many quadratic points if and only if P ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If we just assume that P is a quadratic portrait (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', not necessarily generic), then X1(P) has infinitely many quadratic points if and only if P is a subportrait of some portrait in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' This follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1 as well as the fact that for any quadratic portrait P, if we let P′ be the minimal generic quadratic portrait containing P, then X1(P) and X1(P′) are isomorphic over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (See the discussion preceding Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=') One direction of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1 is straightforward: For every portrait P ∈ Γ, the curve X1(P) is described in at least one of the articles [38,43,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' All the curves in those articles have genus at most 2 and at least one rational point, hence have infinitely many quadratic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Thus, we must show that if P is generic quadratic but not in Γ, then X1(P) has only finitely many quadratic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' To help organize the arguments in the rest of this section, we introduce some terminology: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The cycle structure of a portrait P is the nonincreasing sequence of cycle lengths appearing in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Note that the empty portrait has cycle structure ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If K is a quadratic field and c ∈ K, then the cycle structure of G(fc, K) may contain the integer 1 at most twice and each of the integers 2, 3, and 4 at most once;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' for periods 1 and 2 this follows from the fact that a quadratic polynomial can have at most two fixed points and at most one 2-cycle, and for periods 3 and 4 this comes from [10, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' More precisely, the results of [10] imply that the “period at most 4” portion of the cycle structure of G(fc, K) must be (4,1,1), (4,2), or one of the following: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1) ( ), (1,1), (2), (3), (4), (2,1,1), (3,1,1), (3,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Moreover, it follows from [13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='48] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='21]) that no portrait with both a 4-cycle and a 1-cycle (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=', 4-cycle and a 2-cycle) may be realized infinitely often as G(fc, K) for K a quadratic field and c ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For our purposes, therefore, we may exclude the cycle structures (4,1,1) and (4,2) from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' By enumerating generic quadratic portraits with few vertices, one can verify that if P is a generic quadratic portrait which is not in Γ, then P has a cycle of length n ≥ 5 or P properly contains a portrait in Γ1 or Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We handle these two possibilities separately, showing in each case that the dynamical modular curve X1(P) has only finitely many quadratic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Points of period n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' If P is a generic portrait with a cycle of length n, then there is a dominant morphism X1(P) → X1(n) defined over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In particular, every quadratic point on X1(P) maps to a rational or quadratic point on X1(n), so we need only show that if n ≥ 5, then X1(n) has only finitely many points defined over quadratic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' this is the content of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For n ≥ 1, the cyclic group Cn acts on X1(n) as follows: Given a point (c, z) ∈ X1(n), we also have σn(c, z) := (c, fc(z)) ∈ X1(n), so σ defines an order-n automorphism of X1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' We denote by πn : X1(n) → X0(n) the quotient of X1(n) by this cyclic group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' (The curve X0(n) parametrizes maps fc together with a marked cycle of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=') Let c1,n and c0,n denote the maps from X1(n) and X0(n), respectively, to the c-line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' note that c1,n = c0,n ◦ πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For a curve X, we will denote by gX its genus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' for simplicity, for each n ≥ 1 we will write g1,n and g0,n for the genera of X1(n) and X0(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Finally, recall that we denote by D(n) the degree (in z) of the polynomial Φn(c, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' a formula for D(n) is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='1), and using that formula one can show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='2) 2n−1 ≤ D(n) ≤ 2n, with equality on the right if and only if n = 1 and on the left if and only if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' For all n ≥ 5, we have g0,n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' In [37, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 13], Morton gives an explicit formula for g0,n as well as the lower bound g0,n ≥ 3 2 + �1 4 − 1 n � 2n − (n + 1)2n/2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Rewriting the right-hand expression and using the assumption that n ≥ 5, we have g0,n ≥ 3 2 + 2n/2−1 ��1 4 − 1 5 � 2n/2+1 − (n + 1) � = 3 2 + 2n/2−1 � 1 102n/2 − (n + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The expression � 1 102n/2 − (n + 1) � is positive for all n ≥ 15, so for all such n we have g0,n > 3/2, hence g0,n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Finally, using the explicit formula for g0,n given by Morton, we can exactly compute g0,n for all 1 ≤ n ≤ 14, and we find that g0,n < 2 if and only if 1 ≤ n ≤ 4, in which case g0,n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Let n ≥ 6, and let Rn be the ramification divisor of πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Then deg Rn > 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' It suffices to replace Rn with R0 n, the restriction of the ramification divisor to points that do not map to ∞ under c1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' The ramification divisors of the maps c1,n and c0,n are explicitly computed by Morton in [37, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content=' 11, 13], from which it follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfofjr/content/2301.00510v1.pdf'} +page_content='3) deg R0 n = 1 2 � d|n d 0, Eq. (5) provides an upper +bound for |C(0) − C(t)|: +|C(0) − C(t)| ≤ 2S2 +max sin +� +1 +2 +� t +0 +� +A(t′) +t′ +dt′ +� +, +(6) +where 0 ≤ +1 +2 +� t +0 +√ +A(t′) +t′ +dt′ ≤ +π +2 . Moreover, let ϵ be an +infinitesimally small positive value. Substituting t1 = t− +ϵ and t2 = t into Eq. (5) and using the Taylor expansion +to the sinusoidal function, we obtain +���� +dC(t) +dt +���� ≤ S2 +max +� +A(t) +t +. +(7) +Equation (7) states that the absolute change of the corre- +lation function is determined by the dynamical activity. +Equation (6) holds for 0 ≤ +1 +2 +� t +0 +√ +A(t′) +t′ +dt′ ≤ +π +2 and +thus the predictive power of the bound is lost at a finite +time. An alternative bound to Eq. (5) is given by +|C(0) − C(t)| ≤ 2S2 +max +� +1 − η(t), +(8) +where η(t) is the Loschmidt echo [65] between time +evolved state and the initial state in the continuous + +3 +(a) +(b) +FIG. 2. +Results of numerical simulations. +(a) The ratio +|∂tC(t)|/(S2 +max +� +A(t)/t) for the dichotomous process. +The +result obtained with W12 = 1, W22 = −1, P(0) = [0, 1] is plot- +ted by the dashed line. The results obtained by random pa- +rameters are plotted by the solid lines. The parameter ranges +for the random realizations are W12 ∈ [0, 1], W21 ∈ [0, 1], +S(B1) ∈ [−1, 0], and S(B2) ∈ [0, 1]. +The initial distribu- +tion is first sampled from P1(0) ∈ [0, 1] and P2(0) ∈ [0, 1] +and then normalize the sampled distribution. +(b) The ra- +tio |∂tC(t)|/(S2 +max +� +B(t)/t) for the driven two level atom +model. The results obtained by random parameters are plot- +ted by the solid lines. The parameter ranges are Ω ∈ [0.1, 1], +∆ ∈ [0.1, 1], and κ ∈ [0.1, 1]. The initial density is sampled +from ⟨g|ρ(0)|g⟩ ∈ [0, 1] and ⟨e|ρ(0)|e⟩ ∈ [1, 2] and normalized +the sampled density (non-diagonal elements are zero). +matrix product state representation (see Appendix C). +Equation (8) is the second result of this paper, whose +proof is provided in Appendix E. Following Ref. [66], we +can compute η(t) for the classical Markov process as fol- +lows: +η(t) ≡ +�� +µ +P(µ; 0) +� +e−t � +ν(̸=µ) Wνµ +�2 +, +(9) +which can be represented by quantities of the Markov +process. Note that the Loschmidt echo η(t) constitutes a +lower bound in a quantum and classical thermodynamic +uncertainty relation [66]. +The term within the square +root in η(t) represents the survival probability that there +is no jump starting from the state Bµ. Therefore, when +the activity of dynamics is lower, the survival probabil- +ity becomes higher and in turn η(t) yields a higher value. +Although the Loschmidt echo η(t) has fewer physical in- +terpretations than dynamical activity, it has the advan- +tage over Eq. (5) that the bound of Eq. (8) holds for any +value of t. +We can extend Eq. (5) to a quantum Markov pro- +cess. +Let ρ(t) be a density operator of a quantum +Markov process at time t. We assume that the dynamics +of ρ(t) is governed by the following Lindblad equation +˙ρ(t) = L(ρ(t)), where L is the Lindblad superoperator +[67, 68]: +L(ρ(t)) ≡ −i [H, ρ(t)] + +� +m +D (ρ(t), Lm) . +(10) +Here H is a Hamiltonian, D(ρ, L) ≡ LρL† − {L†L, ρ}/2 +is the dissipator, Lm is the mth jump operator. +We +can unravel Eq. (10) to obtain a quantum trajectory, +which is a measurement record when observing the en- +vironment. The dynamics of the quantum trajectory is +represented by a stochastic Schr¨odinger equation. Simi- +lar to the classical case, we assign the score function to +the quantum state ρ(t) via S(ρ(t)) = Tr[ρ(t)O], where +O is an Hermitian operator. +Figure 1(b) illustrates +an example of a quantum trajectory, which consists of +continuous state change by the effective Hamiltonian +Heff ≡ H − (i/2) � +m L† +mLm and discontinuous jumps +by Lm. Let ρ(0) be the initial density operator. Then +the correlation function C(t) is calculated by +C(t) = S(ρ(0))S(ρ(t)). +(11) +For 0 ≤ t1 < t2, the following relation holds: +|C(t1) − C(t2)| ≤ 2S2 +max sin +� +1 +2 +� t2 +t1 +� +B(t) +t +dt +� +, +(12) +which holds for 0 ≤ 1 +2 +� t2 +t1 +√ +B(t) +t +dt ≤ π +2 . Here B(t) is the +quantum dynamical activity defined in Ref. [64], which +is the quantum generalization of Eq. (4) (see Eq. (F2) in +Appendix F). B(t) is defined through the quantum Fisher +information. The quantum dynamical activity plays an +important role in a speed limit and a thermodynamic +uncertainty relation [64]. The proof of Eq. (12) is shown +in Appendix E. Equation (12) is the same as Eq. (5) +except that A(t) in Eq. (5) is replaced by its quantum +counter part B(t). Following the same procedure as in +Eq. (7), we obtain +���� +dC(t) +dt +���� ≤ S2 +max +� +B(t) +t +. +(13) +Moreover, the bound of Eq. (8) also holds for the quan- +tum Markov process, where the Loschmidt echo for the +quantum case becomes [66] (Appendix C): +η(t) = +��Tr +� +e−iHefftρ(0) +���2 . +(14) +The Loschmidt echo shown in Eq. (14) also constitutes +the lower bound in a quantum thermodynamic uncer- +tainty relation [66]. +Numerical simulations.—We perform numerical sim- +ulations to verify the correlation bounds [Eqs. (7) and +(13)]. +We first demonstrate Eq. (7) with the classical +dichotomous process [69], which takes only two states +B = {B1, B2}. The dichotomous process has a number of +applications in communication engineering and physics. +We are interested in the ratio between the left and right +hand sides of Eq. (7), i.e., |∂tC(t)|/(S2 +max +� +A(t)/t) which +must be no larger than 1 according to Eq. (7). We set the +score function to S(B1) = −1 and S(B2) = 1, in which +Smax = 1. The transition rate is set to W12 = 1 and +W22 = −1, where the other elements are set to 0. The + +4 +initial distribution is P(0) = [0, 1]. +We plot the ratio +as a function of t in Fig. 2(a) with the dashed line. We +also randomly determine the score function S(Bi), the +transition rate Wnm, and the initial distribution P(0) +and calculate the ratio. The ratio as a function of t for +the random realizations is plotted by the solid line in +Fig. 2(a) (the parameter ranges are shown in the cap- +tion of Fig. 2(a)). We see that all the results are below +1 (the dotted line), which numerically verifies the bound +of Eq. (7). +Next, we consider a simple two-level atom driven by +a classical laser field to check the correlation bound of +Eq. (13), whose dynamics is represented by a Lindblad +equation: H = ∆ |e⟩ ⟨e| + Ω +2 (|e⟩ ⟨g| + |g⟩ ⟨e|) and L = +√κ |g⟩ ⟨e|, where ∆, Ω, and κ are model parameters, and +|e⟩ and |g⟩ are the excited and ground states, respectively. +For the score function, we choose S(ρ) = 2Tr[ρ |e⟩ ⟨e|]−1, +which ranges within [−1, 1] and thus Smax = 1. We calcu- +late |∂tC(t)|/(S2 +max +� +B(t)/t), which is the ratio between +the left and right-hand side of Eq. (13). We randomly de- +termine the model parameters and the initial state (the +parameter ranges are shown in the caption of Fig. 2(b)). +The random realizations are shown by the solid lines in +Fig. 2(b). +Different from the classical case, the corre- +lation oscillates due to the contribution of the effective +Hamiltonian Heff. Since all the random realizations are +below 1 (the dotted line), we numerically verify Eq. (13). +Linear response.—The correlation function C(t) is +closely related to the linear response theory [59]. Here, +we apply the correlation bound of Eqs. (5) and (7) to +the linear response theory (see Appendix G for details). +Suppose that the Markov process is in the steady state +Pst = [Pst(1), . . . , Pst(N)], that satisfies WPst = 0. We +apply a weak perturbation χFf(t) to the master equa- +tion of Eq. (1), that is W → W + χFf(t) in Eq. (1), +where 0 < χ ≪ 1 and F is an N × N matrix, and f(t) is +arbitrary real function of time t. We expand the proba- +bility distribution as P(t) = Pst +χP1(t), where P1(t) is +the first-order correction to the probability distribution. +Collecting the first-order contribution O(χ) in Eq. (1), +P1(t) is given by +P1(t) = +� t +−∞ +eW(t−t′)FPstf(t′)dt′. +(15) +Let us consider a scoring function G(Bn), which may +be different from S(Bn) at the moment, and define the +expectation of G(Bn) by +⟨G⟩ = +� +n +G(Bn)Pst(n) = 1GPst, +(16) +where G ≡ diag[G(B1), . . . , G(BN)]. The change in ⟨G⟩ +due to the perturbation, represented by ∆G ≡ 1GP(t)− +1GPst, is +∆G(t) = χ +� ∞ +−∞ +RG(t − t′)f(t′)dt′, +(17) +where RG(t) is the linear response function: +RG(t) = +� +1GeWtFPst +t ≥ 0 +0 +t < 0 . +(18) +In the linear response regime, any input-output relation +can be expressed through RG(t). From Eq. (3), the time +derivative of C(t) reads ∂tC(t) = 1SeWtWSPst. Com- +paring Eq (18) and ∂tC(t), when G = S and F = WS, +∂tC(t) gives the linear response function of Eq. (18), +which is the statement of the fluctuation-dissipation the- +orem. +As a particular case, let us consider the pulse pertur- +bation, f(t) = δ(t), where δ(t) is the Dirac delta func- +tion. This perturbation corresponds to the application of +a sharp pulsatile perturbation at t = 0. Then the change +of the expectation of S(Bn) under the perturbation F = +WS, represented by ∆S(p), is ∆S(p)(t) = χ∂tC(t) (the +superscript (p) represents that it is the pulse response). +The correlation bound of Eq. (7) gives +���∆S(p)(t) +��� ≤ χS2 +max +� +a +t , +(19) +where +a +is +the +rate +of +dynamical +activity +a +≡ +� +ν,µ,ν̸=µ Pst(µ)Wνµ (note that A(t) = at for a steady +state). Equation (19) relates the dynamical activity with +the effect of the pulse perturbation on the Markov pro- +cess. A step response can be calculated in a similar way. +We apply a constant perturbation switched on at t = 0, +which can be modeled by f(t) = Θ(t) with Θ(t) being +the Heaviside step function. From Eq. (17), we obtain +∆S(s)(t) = χ +� t +0 +RS(t′)dt′. +(20) +Equation (20) leads to the following bound: +|∆S(s)(t)| ≤ 2χS2 +max sin +�√ +at +� +, +(21) +which holds for 0 ≤ +√ +at ≤ π/2. For t outside this range, +the trivial inequality |∆S(s)(t)| ≤ 2χS2 +max holds. +Generalizations.—So far we have been concerned with +the two-point correlation function. It is straightforward +to extend the bounds to the multi-point correlation func- +tions. Let us consider a J-point correlation function: +⟨S(t1)S(t2) · · · S(tJ)⟩ +≡ +� +S(Bn1)S(Bn2) · · · S(BnJ)P(n1; t1) +× P(n2; t2|n1; t1) · · · P(nJ; tJ|nJ−1; tJ−1), +(22) +where 0 ≤ t1 < t2 < · · · < tJ. We can obtain analogous +relations of Eqs. (6) and (8) for Eq. (22). +Markov processes are often represented by multiple +variables. +For example, in stochastic thermodynam- +ics, a multipartite process can reveal the relation be- +tween dissipated heat and information flow [70, 71]. For + +5 +simplicity, here we consider a bivariate Markov pro- +cess defined in (X(t), Y (t)), {(X(t), Y (t))|t ≥ 0} that +satisfies in X(t) ∈ BX and Y (t) ∈ BY . +Moreover, +we define different score functions for X(t) and Y (t), +which are expressed by SX(·) and SY (·), respectively, +and define SX,max ≡ maxB∈BX SX(B) and SY,max ≡ +maxB∈BY SY (B). We are often interested in the correla- +tion CXY (t) ≡ ⟨SX(t)SY (0)⟩. Then, |CXY (0) − CXY (t)| +obeys the same upper bounds of Eqs. (5) and (8) except +that S2 +max is replaced by SX,maxSY,max, which gives a +bound that is tighter than or equal to Eqs. (5) and (8). +Conclusion.—In this Letter, we present a relation be- +tween the correlation function and dynamical activity in +classical and quantum Markov processes. The obtained +bounds hold for arbitrary time-independent transition +rate starting from an arbitrary initial distribution. By +applying the obtained bounds to the linear response the- +ory, we demonstrate that the effect of perturbations on +a steady state system is bounded by the dynamical ac- +tivity. We expect that our findings have the potential to +enhance our understanding of nonequilibrium dynamics, +as the correlation function plays a fundamental role in +thermodynamics. +Appendix A: Continuous matrix product state +The derivation of the correlation bounds employ the +continuous matrix product state [56, 57], which bridges +the quantum field and the stochastic process. The con- +tinuous matrix product state is a type of tensor net- +work representation that is used to describe many-body +quantum systems. In one direction, quantum field states +are analyzed via the corresponding continuous measure- +ment problem. In the opposite direction, the continuous +matrix product state can map a classical or quantum +Markov process into a quantum field so that we can an- +alyze trajectory information from the view point of the +quantum field. +We consider a Lindblad equation [Eq. (10)]. The clas- +sical Markov process given by Eq. (1) can be covered by +the Lindblad equation by setting H = 0 and the jump +operator to be of the form Lm = Lνµ = +� +Wνµ |Bν⟩ ⟨Bµ|, +where {|Bν⟩}ν constitutes the orthonormal basis, corre- +sponding to the classical states B = {Bν}ν, and Wνµ is +the transition rate from |Bµ⟩ to |Bµ⟩. +Applying the continuous measurement on the Lindblad +equation [Eq. (10)], we obtain a trajectory Γ, which is a +record of the measurement, as follows: +Γ ≡ [(t1, m1), (t2, m2), . . . , (tK, mK)], +(A1) +where K is the number of total jumps, tk and mk are +time and type of the kth jump event, respectively. The +evolution of ρ(t) in a given trajectory Γ is governed +by a stochastic Schr¨odinger equation. +By taking the +average of all possible measurements in the stochastic +Schr¨odinger equation, we can recover the original Lind- +blad equation of Eq. (10). +Applying continuous measurement, we obtain a par- +ticular trajectory Γ. In the continuous matrix product +state, such a trajectory is recorded in the following state: +|Γ⟩ ≡ φ† +mK(tK) · · · φ† +m2(t2)φ† +m1(t1) |vac⟩ , +(A2) +where φ(t) is the field operator that satisfies the commu- +tation relation [φm(t), φ† +m′(t′)] = δmm′δ(t − t′), and |vac⟩ +is the vacuum state of φm(t), where φ† +m(t) creates a mth +particle at t. The time evolution of the system and field +state |Γ⟩ is given by +|Φ(t)⟩ = U(t; H, {Lm}) |Φ(0)⟩ , +(A3) +where U(t; H, {Lm}) is given by +U(t; H, {Lm}) ≡ T exp +� +−i +� t +0 +ds (Heff ⊗ IF ++ +� +m +iLm ⊗ φ† +m(s)) +� +, +(A4) +In Eq. (A4), the initial state is represented by |Φ(0)⟩ = +|ψ(0)⟩ ⊗ |vac⟩, with |ψ(0)⟩ being the initial state of the +system; T is the time ordering operator, and IF is the +identity operator in the field. |Φ(t)⟩ records the jump +events occurring within the interval [0, t]. The continuous +matrix product state |Φ(t)⟩ comprises the system, which +corresponds to the state of the Markov process, and the +field, which records jump events. The time of the original +Lindblad equation is expressed by t while that of the +continuous matrix product state is by t. All information +about measurement is recorded by creating particles in +the quantum field through the application of an operator +φ† +m(t) to the vacuum state |vac⟩. +For a small time increment ∆t, considering the time +evolution Eq. (A3) and tracing over the field, the time +evolution of the system is given by the Kraus represen- +tation: +ρ(t + ∆t) = +� +m +Vmρ(t)V † +m, +(A5) +where Vm are Kraus operators: +V0 ≡ I − i∆tH, +(A6) +Vm ≡ +√ +∆tLm +(1 ≤ m). +(A7) +Dividing the interval [0, t] into Z ≫ 1 equipartitioned +intervals, the time evolution from t = 0 to t can be rep- +resented by successive applications of Eq. (A5): +ρ(t) = +� +mZ +· · · +� +m1 +VmZ · · · Vm1 |ψ(0)⟩ ⟨ψ(0)| V † +m1 · · · V † +mZ. +(A8) +Using the continuous matrix product state, we can obtain +all the information about the Markov processes. Given +the initial state |ψ(0)⟩, the trajectory probability within +[0, t] can be obtained via +P(Γ, t) = ⟨Φ(t)|IS ⊗ |Γ⟩ ⟨Γ| |Φ(t)⟩ . +(A9) + +6 +The system state ρ(t) can be computed as follows: +ρ(t) = TrF [|Φ(t)⟩ ⟨Φ(t)|] , +(A10) +where TrF denotes the trace operation with respect to +the field state. +Next, we explain a scaled continuous matrix product +state, which was recently introduced in Ref. [64]. +We +want to study the time evolution of the continuous ma- +trix product state. +Initially, we might consider using +the unitary operator defined in Eq. (A4) as the time- +evolution operator. However, this approach has a prob- +lem when we try to calculate the fidelity between two +continuous matrix product states at different times, be- +cause the integration ranges for |Φ(t1)⟩ and |Φ(t2)⟩ are +different. Therefore, we instead use the scaled represen- +tation. Let us define τ > 0, which is the final time of +the evolution. For 0 ≤ t ≤ τ, the scaled matrix product +state representation is given by +|Ψ(t)⟩ = U +� +τ; t +τ H, +�� +t +τ Lm +�� +|Ψ(0)⟩ , +(A11) +where |Ψ(0)⟩ = |ψ(0)⟩ ⊗ |vac⟩. Here, |Φ(t)⟩ and |Ψ(t)⟩ +represent the states of the genuine and scaled continuous +matrix product states, respectively. In the scaled con- +tinuous matrix product state [Eq. (A11)], H and Lm are +scaled as (t/τ)H and +� +t/τLm, respectively, which corre- +sponds to the Lindblad equation that generates dynamics +that are t/τ times as fast as that of the original dynam- +ics. The scaling allows us to have the same integration +range for all values of t, making it possible to evaluate +the fidelity at different times, that is ⟨Ψ(t2)|Ψ(t1)⟩. As +mentioned above, since the scaled matrix product state +is the same as the original one except for their time scale, +both states provide us with the same information except +for the time scale. At the final time τ, both the origi- +nal and the scaled representations give the same state, +|Φ(τ)⟩ = |Φ(τ)⟩. +Moreover, |Ψ(0)⟩ corresponds to the +null dynamics, that is, the dynamics without any state +change. For instance, the system state can be obtained +by +ρ(t) = TrF [|Ψ(t)⟩ ⟨Ψ(t)|] = TrF [|Φ(t)⟩ ⟨Φ(t)|] . +(A12) +When deriving the correlation bounds, we employ the +scaled representation. +Appendix B: Initially mixed state case +The continuous matrix product state given by Eq. (A3) +only considers initially pure state |ψ(0)⟩. Let us consider +the initially mixed state case. Let ρ(0) be the initial den- +sity operator, which is mixed in general. Let us consider +the ancilla A that purifies ρ(0), that is +ρ(0) = TrA[| ˜ψ(0)⟩ ⟨ ˜ψ(0)|], +(B1) +where TrA is the trace operation with respect to the an- +cilla A. Let us introduce the continuous matrix product +state operator corresponding to Eq. (A4), that is applied +to the purified state: +˜U(t; H, {Lm}) ≡T exp +� +−i +� t +0 +ds(Heff ⊗ IA ⊗ IF ++ +� +m +iLm ⊗ IA ⊗ φ† +m(s)) +� +, +(B2) +The Kraus operators corresponding to Eq. (B2) is given +by +˜Vm = Vm ⊗ IA, +(B3) +where IA is the identity operation in the ancilla and Vm +are defined in Eqs. (A6) and (A7). Using Eq. (B3), it +can be confirmed that the one-step evolution yields +TrA +�� +m +˜Vm |˜Ψ(0)⟩ ⟨˜Ψ(0)| ˜V † +m +� += +� +m +VmTrA +� +|˜Ψ(0)⟩ ⟨˜Ψ(0)| +� +V † +m += +� +m +Vmρ(0)V † +m, +(B4) +which actually yields the consistent time evolution. +Appendix C: Fidelity calculation of continuous +matrix product states +The bounds considered in this Letter relate to the cal- +culation of the quantum Fisher information. Specifically, +we need to calculate the following fidelity: +⟨Ψ(t2)|Ψ(t1)⟩ = TrSF [|Ψ(t1)⟩ ⟨Ψ(t2)|] += TrS [ζ(τ; t1, t2)] , +(C1) +where ζ(τ; t1, t2) ≡ TrF [|Ψ(t1)⟩ ⟨Ψ(t2)|]. ζ(τ; t1, t2) sat- +isfies the two-sided Lindblad equation [72, 73]: +d +dtζ(t; t1, t2) = −iH1ζ + iζH2 + +� +m +L1,mζL† +2,m +− 1 +2 +� +m +� +L† +1,mL1,mζ + ζL† +2,mL2,m +� +, +(C2) +where H1 ≡ (t1/τ)H and L1,m ≡ +� +t1/τLm (H2 and +L2,m are defined in a similar manner). +Note that +ζ(τ; t1, t2) is not a density operator, since its trace is not +necessarily equal to unity. To calculate the fidelity using +Eq. (C2), we solve Eq. (C2) from t = 0 to t = τ with the +initial value ζ(0; t1, t2) = ρ(0). +Using Eq. (C2), we can compute the fidelity between +two scaled continuous matrix product states: +η(τ) ≡ |⟨Ψ(τ)|Ψ(0)⟩|2 +(C3) + +7 +From Eq. (C2), |ζ(t; τ, 0)|2 = η(τ) obeys the following +equation [66]: +˙ζ = −iHeffζ = −iHζ − 1 +2 +� +m +L† +mLmζ. +(C4) +Then the fidelity is obtained as follows: +η(τ) = +��TrS +� +e−iHeffτρ(0) +���2 . +(C5) +The classical case can be calculated by setting H = 0 +[66]. +Appendix D: Derivation of Eq. (5) +Here we provide the derivation of Eq. (5). +Using +the scaled continuous matrix product state, a classical +Markov process can be analyzed via quantum mechan- +ics, and thus we can take advantage of inequalities in +quantum information. Let O be an arbitrary Hermitian +operator, and ⟨O⟩t ≡ Tr[ρ(t)O]. +In the field of quan- +tum speed limit, the following relation was recently used +[25, 26]: +��⟨O⟩t2 − ⟨O⟩t1 +�� = Tr [O(ρ(t2) − ρ(t1))] +≤ ∥O∥op ∥ρ(t2) − ρ(t1)∥tr += 2 ∥O∥op TD (ρ(t2), ρ(t1)) . +(D1) +The second line of Eq. (D1) is due to the H¨older inequal- +ity (see Eq. (H6)). We will use Eq. (D1) for the deriva- +tion. The sketch of the proof for Eq. (5) is as follows: +• Consider the scaled continuous matrix product +state for ρ(t) +• Assign the Hermitian operator that calculates the +correlation function for O +• Obtain an upper bound for the trace distance +TD (ρ(t2), ρ(t1)) using the dynamical activity +When considering classical probability and quantum +spaces in Eq. (D1), Eq. (D1) leads to the classical and +quantum bounds, respectively. +We consider Eq. (D1) for the classical probability +space. Let us assume that two density operators ρ and σ +only have diagonal elements: +ρ = +� +x +p(x) |x⟩ ⟨x| , +(D2) +σ = +� +x +q(x) |x⟩ ⟨x| , +(D3) +where p(x) and q(x) are arbitrary probability distribu- +tions and {|x⟩}x constitutes the orthonormal basis. By +calculating the trace distance [Eq. (H7)] for Eqs. (D2) +and (D3), TD(ρ, σ) reduces to the total variation dis- +tance [Eq. (H12)]: +TD(ρ, σ) = TVD(p, q). +(D4) +Now we consider a particular probability distribution. +The probability of measuring a trajectory Γ and Bν at +the end time is +P(Γ, ν, t) ≡ ⟨Ψ(t)|(|Bν⟩ ⟨Bν| ⊗ |Γ⟩ ⟨Γ|)|Ψ(t)⟩ , +(D5) +where |Ψ(t)⟩ is the scaled continuous matrix product +state. When considering initially mixed state, we may +use |˜Ψ(t)⟩. +Because arccos Bhat(·, ·) constitutes the +geodesic distance under the Fisher information metric +[74], the following relation holds [64]: +1 +2 +� t2 +t1 +� +A(t) +t +dt ≥ arccos [Bhat (P(Γ, ν, t1), P(Γ, ν, t2))] , +(D6) +which yields +cos +� +1 +2 +� t2 +t1 +� +A(t) +t +dt +� +≤ Bhat (P(Γ, ν, t1), P(Γ, ν, t2)) , +(D7) +for 0 ≤ 1 +2 +� t2 +t1 +√ +A(t) +t +dt ≤ π +2 . Substituting Eq. (D7) into +Eq. (H17) to obtain +TVD(P(Γ, ν, t1), P(Γ, ν, t2)) +≤ +� +1 − Bhat (P(Γ, ν, t1), P(Γ, ν, t2))2 +≤ +� +� +� +�1 − cos +� +1 +2 +� t2 +t1 +� +A(t) +t +dt +�2 += sin +� +1 +2 +� t2 +t1 +� +A(t) +t +dt +� +. +(D8) +From Eqs. (D1), (D4), and (D8), we obtain +��⟨O⟩t2 − ⟨O⟩t1 +�� +≤ 2 ∥O∥op sin +� +1 +2 +� t2 +t1 +� +A(t) +t +dt +� +, +(D9) +which holds for 0 ≤ 1 +2 +� t2 +t1 +√ +A(t) +t +dt ≤ π +2 . Equation (D9) +is the central inequality for deriving the thermodynamic +correlation inequality. +We now implement the correlation calculation C(τ) = +⟨S(0)S(τ)⟩ with an Hermitian operator acting on the +scaled continuous matrix product state. Given a trajec- +tory Γ and the final state Bν, we can calculate the cor- +relation S(0)S(τ) using |Ψ(τ)⟩. We assume that a real +function M(Γ, ν) calculates the correlation given such in- +formation: +M(Γ, ν) ≡ S(X(0))S(X(τ)) = S(0)S(τ). +(D10) +Now we introduce an Hermitian operator M, whose +eigendecomposition reads +M = +� +Γ,ν +M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| . +(D11) + +8 +Since Eq. (D11) is the eigendecomposition of M, from +Eq. (H2), the operator norm of M is +∥M∥op = max +Γ,ν M(Γ, ν) += +max +Bi,Bj∈B [S(X(0) = Bi)S(X(τ) = Bj)] += S2 +max, +(D12) +where Smax is the maximum absolute value of S(Bi) for +Bi ∈ B defined in Eq. (2). +When we evaluate M in +|Ψ(τ)⟩, it gives +⟨Ψ(τ)|M|Ψ(τ)⟩ = ⟨Ψ(τ)| +� +Γ,ν +M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| |Ψ(τ)⟩ += +� +Γ,ν +M(Γ, ν)P(Γ, ν, τ) += ⟨S(0)S(τ)⟩ . +(D13) +Because |Ψ(0)⟩ corresponds to the null dynamics (the +state does not evolve at all), ⟨Ψ(0)|M|Ψ(0)⟩ = ⟨S(0)2⟩. +In a similar way, when we consider |Ψ(t)⟩ for 0 < t < +τ, we have ⟨Ψ(t)|M|Ψ(t)⟩ = ⟨S(0)S(t)⟩. +Substituting +Eqs. (D12) and (D13) into Eq. (D9), we finally obtain +Eq. (5) in the main text. +Appendix E: Derivation of Eqs. (8) and (12) +In this section, we derive Eqs. (8) and (12). We evalu- +ate TD(ρ(τ), ρ(0)) in Eq. (D1). Since continuous matrix +product states are pure, we have [Eq. (H10)] +TD(|Ψ(t1)⟩ , |Ψ(t2)⟩) = +� +1 − | ⟨Ψ(t2)|Ψ(t1)⟩ |2. +(E1) +As explained in Appendix C, the fidelity can be com- +puted, which leads to Eq. (8) in the main text. +The quantum case can be derived in a similar manner. +As explained in Eqs. (D10) and (D11), the correlation +function can be computed given a trajectory Γ for the +quantum case as well. +Then, the quantum version of +Eq. (8) is obtained in the same way as the classical bound. +We next derive Eq. (12). Since the Bures angle con- +stitutes the geodesic length under the quantum Fisher +information metric [6, 75], similar to Eq. (D6), the fol- +lowing inequality holds [64]: +arccos |⟨Ψ(t2)|Ψ(t1)⟩| ≤ 1 +2 +� t2 +t1 +� +B(t) +t +dt, +(E2) +where B(t) is the quantum dynamical activity [64] (Ap- +pendix F). For 0 ≤ 1 +2 +� t2 +t1 +√ +B(t) +t +dt ≤ π +2 , we have +cos +� +1 +2 +� t2 +t1 +� +B(t) +t +dt +� +≤ |⟨Ψ(t2)|Ψ(t1)⟩| . +(E3) +Substituting Eq. (E3) into Eq. (E1), we obtain +TD (|Ψ(t1)⟩ , |Ψ(t2)⟩) ≤ sin +� +1 +2 +� t2 +t1 +� +B(t) +t +dt +� +. +(E4) +From Eqs. (D1) and (E4), we obtain Eq. (12) in the main +text. +Appendix F: Quantum dynamical activity +The quantum dynamical activity B(t) is defined +through the quantum Fisher information [64]. The quan- +tum Fisher information for the scaled continuous matrix +product state is calculated as follows: +J (t) = +8 +∆t2 [1 − | ⟨Ψ(t) | Ψ(t + ∆t)⟩ |], +(F1) +where ∆t is a sufficiently small increment. The fidelity +| ⟨Ψ(t) | Ψ(t + ∆t)⟩ | can be computed by the two-sided +Lindblad equation [Eq. (C2)]. The quantum dynamical +activity is defined by [64] +B(t) ≡ J (t) +t2 . +(F2) +Appendix G: Linear response +Here, we show detailed calculations of the linear re- +sponse theory. Let us consider applying a weak pertur- +bation χFf(t) to the master equation (1). Considering +the perturbation expansion with respect to χ, upto the +first order, the probability distribution is expanded as +P(t) = Pst + χP1(t), +(G1) +where P1(t) is the first-order term. Substituting Eq. (G1) +into Eq. (1), we have +d +dt (Pst + χP1(t)) = (W + χFf(t)) (Pst + χP1(t)) , +(G2) +in which collecting the terms with respect to the order of +χ yields +O(χ0) +d +dtPst = WPst, +(G3) +O(χ1) +d +dtP1(t) = WP1(t) + FPstf(t). +(G4) +The zeroth order equation vanishes in definition since Pst +is assumed to be the stationary solution of Eq. (1). P1(t) +is given by Eq. (15) in the main text. In the main text, +we consider a scoring function G(Bn) and its expecta- +tion given by Eq. (16). The change of ⟨G⟩ due to the +perturbation can be represented by +∆G(t) ≡ χ1GP1(t) += χ +� t +−∞ +1GeW(t−t′)FPstf(t′)dt′ += χ +� ∞ +−∞ +RG(t − t′)f(t′)dt′, +(G5) + +9 +where RG(t) is the linear response function [Eq. (18)]. +From Eq. (3), the time derivative of C(t) reads +d +dtC(t) = 1SeWtWSPst. +(G6) +In the main text, we consider the case G = S and +F = WS, which will be assumed in the following. The +perturbation WS can be expressed by +WS = +� +���� +S11W11 +S22W12 +· · · +SNNW1N +S11W21 +S22W22 +SNNW2N +... +... +... +S11WN1 S22WN2 · · · SNNWNN +� +���� . +(G7) +We immediately obtain +RS(t) = d +dtC(t). +(G8) +For the pulse perturbation, f(t) = δ(t), where δ(t) is +the Dirac delta function, we obtain +∆S(p)(t) = χ +� ∞ +−∞ +RS(t − t′)δ(t′)dt′ += χRS(t) += χ d +dtC(t). +(G9) +Using Eq. (7), we obtain Eq. (19). +Next, we consider the step perturbation, i.e., f(t) = +Θ(t), where Θ(t) is the Heaviside step function: +Θ(t) = +� +0 +(t < 0) +1 +(t ≥ 0) . +(G10) +Then we have +∆S(p)(t) = χ +� ∞ +−∞ +RS(t − t′)Θ(t′)dt′ += χ +� t +0 +RS(t − t′)dt′ += χ +� t +0 +RS(t′)dt′ += χ +� t +0 +dC(t′) +dt′ +dt′ += χ (C(t) − C(0)) , +(G11) +which yields Eq. (21) in the main text. +Appendix H: Norm and distance measures +For readers’ convenience, we here review the norm and +distance measures for quantum and classical systems. Let +A and B be arbitrary Hermitian operators. The Shattan +p-norm is defined by +∥A∥p ≡ +� +Tr +��√ +A2 +�p�� 1 +p = +� +� +� +λ∈evals(A) +|λ|p +� +� +1 +p +. +(H1) +For particular p, we have +∥A∥op = ∥A∥∞ = +max +λ∈evals(A) |λ|, +(H2) +∥A∥tr = ∥A∥1 = Tr +�√ +A2 +� +, +(H3) +∥A∥hs = ∥A∥2 = +� +Tr [A2], +(H4) +where evals(A) gives a set of eigenvalues of A. +Equa- +tions (H2), (H3), and (H4) are referred to as the opera- +tor norm, the trace norm, and the Hilbert-Schmidt norm, +respectively. The H¨older inequality states +|Tr [AB]| ≤ ∥A∥p ∥B∥q . +(H5) +where p and q should satisfy 1/p+1/q = 1. When p = q = +2, Eq. (H5) reduces to the Cauchy-Schwarz inequality. In +particular, we use p = ∞ and q = 1 case: +|Tr [AB]| ≤ ∥A∥op ∥B∥tr . +(H6) +Let us define the trace distance and quantum fidelity: +TD(ρ, σ) ≡ 1 +2 ∥ρ − σ∥1 , +(H7) +Fid(ρ, σ) ≡ +� +Tr +�√ρσ√ρ +�2 +. +(H8) +When considering pure states |ψ⟩ and |φ⟩, the fidelity +reduces to the inner product: +Fid (|ψ⟩ , |φ⟩) = |⟨ψ|φ⟩|2 , +(H9) +TD(|ψ⟩ , |φ⟩) = +� +1 − | ⟨ψ|φ⟩ |2 +(H10) +These two distances are related via +TD(ρ, σ) ≤ +� +1 − Fid(ρ, σ). +(H11) +The equality of Eq. (H11) holds when both ρ and σ are +pure [58]. +Let us introduce related classical probability distance +measures. Let p(x) and q(x) be probability distributions. +The total variation distance and the Hellinger distance +are given, respectively, by +TVD(p, q) ≡ 1 +2 +� +x +|p(x) − q(x)|, +(H12) +Hel2(p, q) ≡ 1 +2 +� +x +�� +p(x) − +� +q(x) +�2 +(H13) += 1 − Bhat(p, q) +(H14) + +10 +where Bhat(p, q) is the Bhattacharyya coefficient: +Bhat (p, q) ≡ +� +x +� +p(x)q(x). +(H15) +Between the total variantion and the Hellinger distances, +the following relations are known to hold [76]: +Hel2(p, q) ≤ TVD(p, q) +(H16) +≤ +� +Hel2(p, q)(2 − Hel2(p, q)) +(H17) +≤ +� +2Hel2(p, q). +(H18) +ACKNOWLEDGMENTS +This work was supported by JSPS KAKENHI Grant +Number JP22H03659. +[1] W. Heisenberg, ¨Uber den anschaulichen inhalt der quan- +tentheoretischen kinematik und mechanik, Z. Phys. 43, +172 (1927). +[2] H. P. Robertson, The uncertainty principle, Phys. Rev. +34, 163 (1929). +[3] L. Mandelstam and I. Tamm, The uncertainty relation +between energy and time in non-relativistic quantum me- +chanics, J. Phys. 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Theory 62, 5973 (2016). + diff --git a/2dE1T4oBgHgl3EQfSANm/content/tmp_files/load_file.txt b/2dE1T4oBgHgl3EQfSANm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cc7a4eae0b48fb4dc48626cc5c3c4441e57df2d --- /dev/null +++ b/2dE1T4oBgHgl3EQfSANm/content/tmp_files/load_file.txt @@ -0,0 +1,962 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf,len=961 +page_content='Thermodynamic Correlation Inequality Yoshihiko Hasegawa∗ Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan (Dated: January 10, 2023) Uncertainty relations place fundamental limits on the operations that physical systems can per- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In this Letter, we obtain uncertainty relations that give bounds for the correlation function, which measures the relationship between a system’s current state and its future state, in both classical and quantum Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The obtained bounds, referred to as thermodynamic cor- relation inequality, state that the change in the correlation function has an upper bound comprising the dynamical activity, a measure of the activity of a Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, applying the ob- tained relation to the linear response function, we show that the effect of perturbation has a bound comprising the dynamical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—Uncertainty relations imply that there are ultimate impossibilities in the physical world that cannot be overcome by any technological advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The most well-known example is the Heisenberg uncertainty relation [1, 2], which establishes a limit on the precision of position-momentum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The quantum speed limit is interpreted as the energy-time uncertainty rela- tion and places a limit on the speed at which the quan- tum state can be changed [3–10] (see [11] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' It has many applications in quantum computation [12], quantum communication [13, 14], and quantum thermo- dynamics [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Recently, the concept of speed limit has also been considered in classical systems [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In par- ticular, the Wasserstein distance can be used to obtain the minimum entropy production required for a stochas- tic process to transform one probability distribution into another [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, the speed limit has been gen- eralized to the time evolution of the observables [23–27], where the speed of the observables is the quantity of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' A closely related principle was recently found in stochastic thermodynamics, which is known as the ther- modynamic uncertainty relation [28–50] (see [51] for a re- view), stating that, for thermodynamic systems, higher accuracy can be achieved at the expense of larger ther- modynamic cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Nowadays, the thermodynamic uncer- tainty relations become a central topic in nonequilibrium thermodynamics, and their importance is also recognized from a practical standpoint because thermodynamic un- certainty relations can be used to infer entropy produc- tion without detailed knowledge on the system [52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the present Letter, we obtain uncertainty relations that confer bounds for the correlation function in classical and quantum Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The correlation function is a statistical measure that quantifies the correlation be- tween the current state of a system and its future or past states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In a Markov process, the correlation function can be used to analyze the dependence of the current state on past states, and to identify any patterns in the system’s ∗ hasegawa@biom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='jp behavior over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We derive the thermodynamic corre- lation inequality stating that the amount of the correla- tion change has an upper bound that comprises the dy- namical activity, which quantifies the activity of a system of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Our derivation is based on the continuous ma- trix product state representation [56, 57], which is a real- ization of the bulk/boundary correspondence in Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' It allows us to represent a classical or quan- tum Markov process by the corresponding quantum field state, where jump events in the Markov process are rep- resented by particle creations in the field state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Since the dynamics of the continuous matrix product state is as- sumed to obey that of quantum mechanics, we can apply the techniques developed in quantum information [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The obtained bound exhibits unexpected generality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' it holds for classical as well as quantum Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, it can be generalized to multi-point correlation functions and multivariate Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The cor- relation function gives the spectral information via the Wiener-Khinchin theorem and plays a fundamental role in the linear response theory [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The linear response function can be represented by a time derivative of the corresponding correlation function, which is the state- ment of the fluctuation-dissipation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Applying the obtained correlation bound to the linear response function, we derive an upper bound to the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—We derive the thermodynamic correlation in- equality for a classical Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' A quantum gen- eralization will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Consider a classical Markov process with N states B ≡ {B1, B2, · · · , BN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let {X(t)|t ≥ 0} be a collection of discrete random vari- ables that take values in B (that is X(t) ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let P(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t) be the probability that X(t) is Bν at time t and Wνµ be the transition rate of X(t) from Bµ to Bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The time evo- lution of P(t) ≡ [P(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , P(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t)]⊤ is governed by the following master equation: dP(t) dt = WP(t), (1) where W ≡ {Wνµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Next, we define the scoring function S(·) that takes a state Bi (i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , N}) and returns a real value of (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' When it is clear from the con- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='03060v1 [quant-ph] 8 Jan 2023 2 State Time State Time +1 1 +1 1 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Illustration of Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (a) Classical Markov process (dichotomous process) two states {B1, B2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The score function is specified by S(B1) = −1 and S(B2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (b) Quantum Markov process (two level atom driven by a classical laser field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The time evolution of the quantum Markov process consists of continuous evolution induced by the effective Hamiltonian Heff and discontinous evolution due to the jump operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The score function is given by S(ρ) = 2Tr[ρ |e⟩ ⟨e|] − 1, in which the ground and excited states give S(|g⟩ ⟨g|) = −1 and S(|e⟩ ⟨e|) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' text, we use the notation S(t) ≡ S(X(t)) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, we define Smax ≡ max Bi∈B |S(Bi)|, (2) which is the maximum absolute value of the score func- tion within B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We are interested in the correlation func- tion C(t) ≡ ⟨S(0)S(t)⟩, where ⟨S(0)S(t)⟩ = � µ,ν S(Bν)S(Bµ)P(µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 0)P(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t|µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 0) = 1SeWtSP(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (3) Here, P(ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t|µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 0) is the conditional probability that X(t) = Bν given X(0) = Bµ, 1 ≡ [1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , 1] is the trace state, and S ≡ diag[S(B1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , S(BN)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The cor- relation function C(t) is widely explored in the field of stochastic process [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Recently, the correlation func- tion is considered in the context of quantum speed limit [26, 62], which is obtained as particular cases of speed limit on observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' As an example of the classical system, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 1(a) shows the dichotomous process, which comprises two states {B1, B2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' X(t) in this process ex- hibits random switching between B1 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For the di- chotomous process, the score function is typically given by S(B1) = −1 and S(B2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' To quantify the Markov process, we define the dynamical activity A(t) as follows [63]: A(t) ≡ � t 0 dt′ � ν,µ,ν̸=µ P(µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t′)Wνµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (4) A(t) represents the average number of jumps during the interval [0, t] and it quantifies the activity of the stochas- tic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The dynamical activity plays a fundamental role in classical speed limits [15] and thermodynamic un- certainty relations [30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the classical Markov process, we obtain an upper bound on the correlation function C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For 0 ≤ t1 < t2, we obtain the following bound: |C(t1) − C(t2)| ≤ 2S2 max sin � 1 2 � t2 t1 � A(t) t dt � , (5) which holds for 0 ≤ 1 2 � t2 t1 √ A(t) t dt ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For t1 and t2 outside this range, the upper bound is |C(t1) − C(t2)| ≤ 2S2 max, which holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (5) is the main result of this Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' It should be emphasized that all the quantities appeared in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) are physically inter- pretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) is based on the continuous matrix product state and inequalities in quantum infor- mation, which is shown in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (5) holds for an arbitrary time-independent Markov process starting from an arbitrary initial probability distribution with an arbitrary score function S(Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (5) states that higher dynamical activity allows the system to forget its current state more quickly, which agrees with our intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For a simple consistency check, consider the null dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=', Wνµ = 0 for all ν and µ), in which there is no jump at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In this case, the dynamical activity becomes A(t) = 0 and thus the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) vanishes to yield ⟨S(0)S(t)⟩ = ⟨S(0)2⟩, which is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For the steady state case, C(t2) − C(t1) for t1 < t2 is negative, and hence it seems that we do not have to consider absolute operation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' However, when the system is not in steady state, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Note that a weaker bound can be obtained via a thermodynamic uncertainty relation derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us consider particular cases of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Simply taking t1 = 0 and t2 = t with t > 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) provides an upper bound for |C(0) − C(t)|: |C(0) − C(t)| ≤ 2S2 max sin � 1 2 � t 0 � A(t′) t′ dt′ � , (6) where 0 ≤ 1 2 � t 0 √ A(t′) t′ dt′ ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, let ϵ be an infinitesimally small positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Substituting t1 = t− ϵ and t2 = t into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) and using the Taylor expansion to the sinusoidal function, we obtain ���� dC(t) dt ���� ≤ S2 max � A(t) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7) Equation (7) states that the absolute change of the corre- lation function is determined by the dynamical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (6) holds for 0 ≤ 1 2 � t 0 √ A(t′) t′ dt′ ≤ π 2 and thus the predictive power of the bound is lost at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' An alternative bound to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) is given by |C(0) − C(t)| ≤ 2S2 max � 1 − η(t), (8) where η(t) is the Loschmidt echo [65] between time evolved state and the initial state in the continuous 3 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Results of numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (a) The ratio |∂tC(t)|/(S2 max � A(t)/t) for the dichotomous process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The result obtained with W12 = 1, W22 = −1, P(0) = [0, 1] is plot- ted by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The results obtained by random pa- rameters are plotted by the solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The parameter ranges for the random realizations are W12 ∈ [0, 1], W21 ∈ [0, 1], S(B1) ∈ [−1, 0], and S(B2) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The initial distribu- tion is first sampled from P1(0) ∈ [0, 1] and P2(0) ∈ [0, 1] and then normalize the sampled distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (b) The ra- tio |∂tC(t)|/(S2 max � B(t)/t) for the driven two level atom model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The results obtained by random parameters are plot- ted by the solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The parameter ranges are Ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='1, 1], ∆ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='1, 1], and κ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The initial density is sampled from ⟨g|ρ(0)|g⟩ ∈ [0, 1] and ⟨e|ρ(0)|e⟩ ∈ [1, 2] and normalized the sampled density (non-diagonal elements are zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' matrix product state representation (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (8) is the second result of this paper, whose proof is provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' [66], we can compute η(t) for the classical Markov process as fol- lows: η(t) ≡ �� µ P(µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 0) � e−t � ν(̸=µ) Wνµ �2 , (9) which can be represented by quantities of the Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Note that the Loschmidt echo η(t) constitutes a lower bound in a quantum and classical thermodynamic uncertainty relation [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The term within the square root in η(t) represents the survival probability that there is no jump starting from the state Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Therefore, when the activity of dynamics is lower, the survival probabil- ity becomes higher and in turn η(t) yields a higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Although the Loschmidt echo η(t) has fewer physical in- terpretations than dynamical activity, it has the advan- tage over Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) that the bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) holds for any value of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We can extend Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) to a quantum Markov pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let ρ(t) be a density operator of a quantum Markov process at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We assume that the dynamics of ρ(t) is governed by the following Lindblad equation ˙ρ(t) = L(ρ(t)), where L is the Lindblad superoperator [67, 68]: L(ρ(t)) ≡ −i [H, ρ(t)] + � m D (ρ(t), Lm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (10) Here H is a Hamiltonian, D(ρ, L) ≡ LρL† − {L†L, ρ}/2 is the dissipator, Lm is the mth jump operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We can unravel Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (10) to obtain a quantum trajectory, which is a measurement record when observing the en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The dynamics of the quantum trajectory is represented by a stochastic Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Simi- lar to the classical case, we assign the score function to the quantum state ρ(t) via S(ρ(t)) = Tr[ρ(t)O], where O is an Hermitian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Figure 1(b) illustrates an example of a quantum trajectory, which consists of continuous state change by the effective Hamiltonian Heff ≡ H − (i/2) � m L† mLm and discontinuous jumps by Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let ρ(0) be the initial density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Then the correlation function C(t) is calculated by C(t) = S(ρ(0))S(ρ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (11) For 0 ≤ t1 < t2, the following relation holds: |C(t1) − C(t2)| ≤ 2S2 max sin � 1 2 � t2 t1 � B(t) t dt � , (12) which holds for 0 ≤ 1 2 � t2 t1 √ B(t) t dt ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Here B(t) is the quantum dynamical activity defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' [64], which is the quantum generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (4) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (F2) in Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' B(t) is defined through the quantum Fisher information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The quantum dynamical activity plays an important role in a speed limit and a thermodynamic uncertainty relation [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (12) is shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (12) is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) except that A(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) is replaced by its quantum counter part B(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Following the same procedure as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7), we obtain ���� dC(t) dt ���� ≤ S2 max � B(t) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (13) Moreover, the bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) also holds for the quan- tum Markov process, where the Loschmidt echo for the quantum case becomes [66] (Appendix C): η(t) = ��Tr � e−iHefftρ(0) ���2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (14) The Loschmidt echo shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (14) also constitutes the lower bound in a quantum thermodynamic uncer- tainty relation [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—We perform numerical sim- ulations to verify the correlation bounds [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7) and (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We first demonstrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7) with the classical dichotomous process [69], which takes only two states B = {B1, B2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The dichotomous process has a number of applications in communication engineering and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We are interested in the ratio between the left and right hand sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=', |∂tC(t)|/(S2 max � A(t)/t) which must be no larger than 1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We set the score function to S(B1) = −1 and S(B2) = 1, in which Smax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The transition rate is set to W12 = 1 and W22 = −1, where the other elements are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The 4 initial distribution is P(0) = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We plot the ratio as a function of t in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2(a) with the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We also randomly determine the score function S(Bi), the transition rate Wnm, and the initial distribution P(0) and calculate the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The ratio as a function of t for the random realizations is plotted by the solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2(a) (the parameter ranges are shown in the cap- tion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We see that all the results are below 1 (the dotted line), which numerically verifies the bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Next, we consider a simple two-level atom driven by a classical laser field to check the correlation bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (13), whose dynamics is represented by a Lindblad equation: H = ∆ |e⟩ ⟨e| + Ω 2 (|e⟩ ⟨g| + |g⟩ ⟨e|) and L = √κ |g⟩ ⟨e|, where ∆, Ω, and κ are model parameters, and |e⟩ and |g⟩ are the excited and ground states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For the score function, we choose S(ρ) = 2Tr[ρ |e⟩ ⟨e|]−1, which ranges within [−1, 1] and thus Smax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We calcu- late |∂tC(t)|/(S2 max � B(t)/t), which is the ratio between the left and right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We randomly de- termine the model parameters and the initial state (the parameter ranges are shown in the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The random realizations are shown by the solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Different from the classical case, the corre- lation oscillates due to the contribution of the effective Hamiltonian Heff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Since all the random realizations are below 1 (the dotted line), we numerically verify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—The correlation function C(t) is closely related to the linear response theory [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Here, we apply the correlation bound of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) and (7) to the linear response theory (see Appendix G for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Suppose that the Markov process is in the steady state Pst = [Pst(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , Pst(N)], that satisfies WPst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We apply a weak perturbation χFf(t) to the master equa- tion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1), that is W → W + χFf(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1), where 0 < χ ≪ 1 and F is an N × N matrix, and f(t) is arbitrary real function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We expand the proba- bility distribution as P(t) = Pst +χP1(t), where P1(t) is the first-order correction to the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Collecting the first-order contribution O(χ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1), P1(t) is given by P1(t) = � t −∞ eW(t−t′)FPstf(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (15) Let us consider a scoring function G(Bn), which may be different from S(Bn) at the moment, and define the expectation of G(Bn) by ⟨G⟩ = � n G(Bn)Pst(n) = 1GPst, (16) where G ≡ diag[G(B1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , G(BN)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The change in ⟨G⟩ due to the perturbation, represented by ∆G ≡ 1GP(t)− 1GPst, is ∆G(t) = χ � ∞ −∞ RG(t − t′)f(t′)dt′, (17) where RG(t) is the linear response function: RG(t) = � 1GeWtFPst t ≥ 0 0 t < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (18) In the linear response regime, any input-output relation can be expressed through RG(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (3), the time derivative of C(t) reads ∂tC(t) = 1SeWtWSPst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Com- paring Eq (18) and ∂tC(t), when G = S and F = WS, ∂tC(t) gives the linear response function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (18), which is the statement of the fluctuation-dissipation the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' As a particular case, let us consider the pulse pertur- bation, f(t) = δ(t), where δ(t) is the Dirac delta func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' This perturbation corresponds to the application of a sharp pulsatile perturbation at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Then the change of the expectation of S(Bn) under the perturbation F = WS, represented by ∆S(p), is ∆S(p)(t) = χ∂tC(t) (the superscript (p) represents that it is the pulse response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The correlation bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7) gives ���∆S(p)(t) ��� ≤ χS2 max � a t , (19) where a is the rate of dynamical activity a ≡ � ν,µ,ν̸=µ Pst(µ)Wνµ (note that A(t) = at for a steady state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (19) relates the dynamical activity with the effect of the pulse perturbation on the Markov pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' A step response can be calculated in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We apply a constant perturbation switched on at t = 0, which can be modeled by f(t) = Θ(t) with Θ(t) being the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (17), we obtain ∆S(s)(t) = χ � t 0 RS(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (20) Equation (20) leads to the following bound: |∆S(s)(t)| ≤ 2χS2 max sin �√ at � , (21) which holds for 0 ≤ √ at ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For t outside this range, the trivial inequality |∆S(s)(t)| ≤ 2χS2 max holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—So far we have been concerned with the two-point correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' It is straightforward to extend the bounds to the multi-point correlation func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us consider a J-point correlation function: ⟨S(t1)S(t2) · · · S(tJ)⟩ ≡ � S(Bn1)S(Bn2) · · · S(BnJ)P(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1) × P(n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t2|n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1) · · · P(nJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' tJ|nJ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' tJ−1), (22) where 0 ≤ t1 < t2 < · · · < tJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We can obtain analogous relations of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (6) and (8) for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Markov processes are often represented by multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For example, in stochastic thermodynam- ics, a multipartite process can reveal the relation be- tween dissipated heat and information flow [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For 5 simplicity, here we consider a bivariate Markov pro- cess defined in (X(t), Y (t)), {(X(t), Y (t))|t ≥ 0} that satisfies in X(t) ∈ BX and Y (t) ∈ BY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, we define different score functions for X(t) and Y (t), which are expressed by SX(·) and SY (·), respectively, and define SX,max ≡ maxB∈BX SX(B) and SY,max ≡ maxB∈BY SY (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We are often interested in the correla- tion CXY (t) ≡ ⟨SX(t)SY (0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Then, |CXY (0) − CXY (t)| obeys the same upper bounds of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) and (8) except that S2 max is replaced by SX,maxSY,max, which gives a bound that is tighter than or equal to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='—In this Letter, we present a relation be- tween the correlation function and dynamical activity in classical and quantum Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The obtained bounds hold for arbitrary time-independent transition rate starting from an arbitrary initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' By applying the obtained bounds to the linear response the- ory, we demonstrate that the effect of perturbations on a steady state system is bounded by the dynamical ac- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We expect that our findings have the potential to enhance our understanding of nonequilibrium dynamics, as the correlation function plays a fundamental role in thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix A: Continuous matrix product state The derivation of the correlation bounds employ the continuous matrix product state [56, 57], which bridges the quantum field and the stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The con- tinuous matrix product state is a type of tensor net- work representation that is used to describe many-body quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In one direction, quantum field states are analyzed via the corresponding continuous measure- ment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the opposite direction, the continuous matrix product state can map a classical or quantum Markov process into a quantum field so that we can an- alyze trajectory information from the view point of the quantum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We consider a Lindblad equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The clas- sical Markov process given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1) can be covered by the Lindblad equation by setting H = 0 and the jump operator to be of the form Lm = Lνµ = � Wνµ |Bν⟩ ⟨Bµ|, where {|Bν⟩}ν constitutes the orthonormal basis, corre- sponding to the classical states B = {Bν}ν, and Wνµ is the transition rate from |Bµ⟩ to |Bµ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Applying the continuous measurement on the Lindblad equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (10)], we obtain a trajectory Γ, which is a record of the measurement, as follows: Γ ≡ [(t1, m1), (t2, m2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' , (tK, mK)], (A1) where K is the number of total jumps, tk and mk are time and type of the kth jump event, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The evolution of ρ(t) in a given trajectory Γ is governed by a stochastic Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' By taking the average of all possible measurements in the stochastic Schr¨odinger equation, we can recover the original Lind- blad equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Applying continuous measurement, we obtain a par- ticular trajectory Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the continuous matrix product state, such a trajectory is recorded in the following state: |Γ⟩ ≡ φ† mK(tK) · · · φ† m2(t2)φ† m1(t1) |vac⟩ , (A2) where φ(t) is the field operator that satisfies the commu- tation relation [φm(t), φ† m′(t′)] = δmm′δ(t − t′), and |vac⟩ is the vacuum state of φm(t), where φ† m(t) creates a mth particle at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The time evolution of the system and field state |Γ⟩ is given by |Φ(t)⟩ = U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' H, {Lm}) |Φ(0)⟩ , (A3) where U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' H, {Lm}) is given by U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' H, {Lm}) ≡ T exp � −i � t 0 ds (Heff ⊗ IF + � m iLm ⊗ φ† m(s)) � , (A4) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A4), the initial state is represented by |Φ(0)⟩ = |ψ(0)⟩ ⊗ |vac⟩, with |ψ(0)⟩ being the initial state of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' T is the time ordering operator, and IF is the identity operator in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' |Φ(t)⟩ records the jump events occurring within the interval [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The continuous matrix product state |Φ(t)⟩ comprises the system, which corresponds to the state of the Markov process, and the field, which records jump events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The time of the original Lindblad equation is expressed by t while that of the continuous matrix product state is by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' All information about measurement is recorded by creating particles in the quantum field through the application of an operator φ† m(t) to the vacuum state |vac⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For a small time increment ∆t, considering the time evolution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A3) and tracing over the field, the time evolution of the system is given by the Kraus represen- tation: ρ(t + ∆t) = � m Vmρ(t)V † m, (A5) where Vm are Kraus operators: V0 ≡ I − i∆tH, (A6) Vm ≡ √ ∆tLm (1 ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A7) Dividing the interval [0, t] into Z ≫ 1 equipartitioned intervals, the time evolution from t = 0 to t can be rep- resented by successive applications of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A5): ρ(t) = � mZ · · � m1 VmZ · · · Vm1 |ψ(0)⟩ ⟨ψ(0)| V † m1 · · · V † mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A8) Using the continuous matrix product state, we can obtain all the information about the Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Given the initial state |ψ(0)⟩, the trajectory probability within [0, t] can be obtained via P(Γ, t) = ⟨Φ(t)|IS ⊗ |Γ⟩ ⟨Γ| |Φ(t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A9) 6 The system state ρ(t) can be computed as follows: ρ(t) = TrF [|Φ(t)⟩ ⟨Φ(t)|] , (A10) where TrF denotes the trace operation with respect to the field state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Next, we explain a scaled continuous matrix product state, which was recently introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We want to study the time evolution of the continuous ma- trix product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Initially, we might consider using the unitary operator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A4) as the time- evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' However, this approach has a prob- lem when we try to calculate the fidelity between two continuous matrix product states at different times, be- cause the integration ranges for |Φ(t1)⟩ and |Φ(t2)⟩ are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Therefore, we instead use the scaled represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us define τ > 0, which is the final time of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For 0 ≤ t ≤ τ, the scaled matrix product state representation is given by |Ψ(t)⟩ = U � τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t τ H, �� t τ Lm �� |Ψ(0)⟩ , (A11) where |Ψ(0)⟩ = |ψ(0)⟩ ⊗ |vac⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Here, |Φ(t)⟩ and |Ψ(t)⟩ represent the states of the genuine and scaled continuous matrix product states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the scaled con- tinuous matrix product state [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A11)], H and Lm are scaled as (t/τ)H and � t/τLm, respectively, which corre- sponds to the Lindblad equation that generates dynamics that are t/τ times as fast as that of the original dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The scaling allows us to have the same integration range for all values of t, making it possible to evaluate the fidelity at different times, that is ⟨Ψ(t2)|Ψ(t1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' As mentioned above, since the scaled matrix product state is the same as the original one except for their time scale, both states provide us with the same information except for the time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' At the final time τ, both the origi- nal and the scaled representations give the same state, |Φ(τ)⟩ = |Φ(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Moreover, |Ψ(0)⟩ corresponds to the null dynamics, that is, the dynamics without any state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For instance, the system state can be obtained by ρ(t) = TrF [|Ψ(t)⟩ ⟨Ψ(t)|] = TrF [|Φ(t)⟩ ⟨Φ(t)|] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A12) When deriving the correlation bounds, we employ the scaled representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix B: Initially mixed state case The continuous matrix product state given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A3) only considers initially pure state |ψ(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us consider the initially mixed state case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let ρ(0) be the initial den- sity operator, which is mixed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us consider the ancilla A that purifies ρ(0), that is ρ(0) = TrA[| ˜ψ(0)⟩ ⟨ ˜ψ(0)|], (B1) where TrA is the trace operation with respect to the an- cilla A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us introduce the continuous matrix product state operator corresponding to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A4), that is applied to the purified state: ˜U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' H, {Lm}) ≡T exp � −i � t 0 ds(Heff ⊗ IA ⊗ IF + � m iLm ⊗ IA ⊗ φ† m(s)) � , (B2) The Kraus operators corresponding to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (B2) is given by ˜Vm = Vm ⊗ IA, (B3) where IA is the identity operation in the ancilla and Vm are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (A6) and (A7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (B3), it can be confirmed that the one-step evolution yields TrA �� m ˜Vm |˜Ψ(0)⟩ ⟨˜Ψ(0)| ˜V † m � = � m VmTrA � |˜Ψ(0)⟩ ⟨˜Ψ(0)| � V † m = � m Vmρ(0)V † m, (B4) which actually yields the consistent time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix C: Fidelity calculation of continuous matrix product states The bounds considered in this Letter relate to the cal- culation of the quantum Fisher information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Specifically, we need to calculate the following fidelity: ⟨Ψ(t2)|Ψ(t1)⟩ = TrSF [|Ψ(t1)⟩ ⟨Ψ(t2)|] = TrS [ζ(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2)] , (C1) where ζ(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2) ≡ TrF [|Ψ(t1)⟩ ⟨Ψ(t2)|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' ζ(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2) sat- isfies the two-sided Lindblad equation [72, 73]: d dtζ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2) = −iH1ζ + iζH2 + � m L1,mζL† 2,m − 1 2 � m � L† 1,mL1,mζ + ζL† 2,mL2,m � , (C2) where H1 ≡ (t1/τ)H and L1,m ≡ � t1/τLm (H2 and L2,m are defined in a similar manner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Note that ζ(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2) is not a density operator, since its trace is not necessarily equal to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' To calculate the fidelity using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C2), we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C2) from t = 0 to t = τ with the initial value ζ(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' t1, t2) = ρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C2), we can compute the fidelity between two scaled continuous matrix product states: η(τ) ≡ |⟨Ψ(τ)|Ψ(0)⟩|2 (C3) 7 From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C2), |ζ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' τ, 0)|2 = η(τ) obeys the following equation [66]: ˙ζ = −iHeffζ = −iHζ − 1 2 � m L† mLmζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C4) Then the fidelity is obtained as follows: η(τ) = ��TrS � e−iHeffτρ(0) ���2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C5) The classical case can be calculated by setting H = 0 [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix D: Derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) Here we provide the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Using the scaled continuous matrix product state, a classical Markov process can be analyzed via quantum mechan- ics, and thus we can take advantage of inequalities in quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let O be an arbitrary Hermitian operator, and ⟨O⟩t ≡ Tr[ρ(t)O].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the field of quan- tum speed limit, the following relation was recently used [25, 26]: ��⟨O⟩t2 − ⟨O⟩t1 �� = Tr [O(ρ(t2) − ρ(t1))] ≤ ∥O∥op ∥ρ(t2) − ρ(t1)∥tr = 2 ∥O∥op TD (ρ(t2), ρ(t1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) The second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) is due to the H¨older inequal- ity (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We will use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) for the deriva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The sketch of the proof for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) is as follows: Consider the scaled continuous matrix product state for ρ(t) Assign the Hermitian operator that calculates the correlation function for O Obtain an upper bound for the trace distance TD (ρ(t2), ρ(t1)) using the dynamical activity When considering classical probability and quantum spaces in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) leads to the classical and quantum bounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) for the classical probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us assume that two density operators ρ and σ only have diagonal elements: ρ = � x p(x) |x⟩ ⟨x| , (D2) σ = � x q(x) |x⟩ ⟨x| , (D3) where p(x) and q(x) are arbitrary probability distribu- tions and {|x⟩}x constitutes the orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' By calculating the trace distance [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H7)] for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D2) and (D3), TD(ρ, σ) reduces to the total variation dis- tance [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H12)]: TD(ρ, σ) = TVD(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D4) Now we consider a particular probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The probability of measuring a trajectory Γ and Bν at the end time is P(Γ, ν, t) ≡ ⟨Ψ(t)|(|Bν⟩ ⟨Bν| ⊗ |Γ⟩ ⟨Γ|)|Ψ(t)⟩ , (D5) where |Ψ(t)⟩ is the scaled continuous matrix product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' When considering initially mixed state, we may use |˜Ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Because arccos Bhat(·, ·) constitutes the geodesic distance under the Fisher information metric [74], the following relation holds [64]: 1 2 � t2 t1 � A(t) t dt ≥ arccos [Bhat (P(Γ, ν, t1), P(Γ, ν, t2))] , (D6) which yields cos � 1 2 � t2 t1 � A(t) t dt � ≤ Bhat (P(Γ, ν, t1), P(Γ, ν, t2)) , (D7) for 0 ≤ 1 2 � t2 t1 √ A(t) t dt ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H17) to obtain TVD(P(Γ, ν, t1), P(Γ, ν, t2)) ≤ � 1 − Bhat (P(Γ, ν, t1), P(Γ, ν, t2))2 ≤ � � � �1 − cos � 1 2 � t2 t1 � A(t) t dt �2 = sin � 1 2 � t2 t1 � A(t) t dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D8) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1), (D4), and (D8), we obtain ��⟨O⟩t2 − ⟨O⟩t1 �� ≤ 2 ∥O∥op sin � 1 2 � t2 t1 � A(t) t dt � , (D9) which holds for 0 ≤ 1 2 � t2 t1 √ A(t) t dt ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equation (D9) is the central inequality for deriving the thermodynamic correlation inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We now implement the correlation calculation C(τ) = ⟨S(0)S(τ)⟩ with an Hermitian operator acting on the scaled continuous matrix product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Given a trajec- tory Γ and the final state Bν, we can calculate the cor- relation S(0)S(τ) using |Ψ(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We assume that a real function M(Γ, ν) calculates the correlation given such in- formation: M(Γ, ν) ≡ S(X(0))S(X(τ)) = S(0)S(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D10) Now we introduce an Hermitian operator M, whose eigendecomposition reads M = � Γ,ν M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D11) 8 Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D11) is the eigendecomposition of M, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H2), the operator norm of M is ∥M∥op = max Γ,ν M(Γ, ν) = max Bi,Bj∈B [S(X(0) = Bi)S(X(τ) = Bj)] = S2 max, (D12) where Smax is the maximum absolute value of S(Bi) for Bi ∈ B defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' When we evaluate M in |Ψ(τ)⟩, it gives ⟨Ψ(τ)|M|Ψ(τ)⟩ = ⟨Ψ(τ)| � Γ,ν M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| |Ψ(τ)⟩ = � Γ,ν M(Γ, ν)P(Γ, ν, τ) = ⟨S(0)S(τ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D13) Because |Ψ(0)⟩ corresponds to the null dynamics (the state does not evolve at all), ⟨Ψ(0)|M|Ψ(0)⟩ = ⟨S(0)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In a similar way, when we consider |Ψ(t)⟩ for 0 < t < τ, we have ⟨Ψ(t)|M|Ψ(t)⟩ = ⟨S(0)S(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D12) and (D13) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D9), we finally obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (5) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix E: Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) and (12) In this section, we derive Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We evalu- ate TD(ρ(τ), ρ(0)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Since continuous matrix product states are pure, we have [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H10)] TD(|Ψ(t1)⟩ , |Ψ(t2)⟩) = � 1 − | ⟨Ψ(t2)|Ψ(t1)⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (E1) As explained in Appendix C, the fidelity can be com- puted, which leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The quantum case can be derived in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' As explained in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D10) and (D11), the correlation function can be computed given a trajectory Γ for the quantum case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Then, the quantum version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (8) is obtained in the same way as the classical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' We next derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Since the Bures angle con- stitutes the geodesic length under the quantum Fisher information metric [6, 75], similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D6), the fol- lowing inequality holds [64]: arccos |⟨Ψ(t2)|Ψ(t1)⟩| ≤ 1 2 � t2 t1 � B(t) t dt, (E2) where B(t) is the quantum dynamical activity [64] (Ap- pendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' For 0 ≤ 1 2 � t2 t1 √ B(t) t dt ≤ π 2 , we have cos � 1 2 � t2 t1 � B(t) t dt � ≤ |⟨Ψ(t2)|Ψ(t1)⟩| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (E3) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (E3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (E1), we obtain TD (|Ψ(t1)⟩ , |Ψ(t2)⟩) ≤ sin � 1 2 � t2 t1 � B(t) t dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (E4) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (D1) and (E4), we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (12) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix F: Quantum dynamical activity The quantum dynamical activity B(t) is defined through the quantum Fisher information [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The quan- tum Fisher information for the scaled continuous matrix product state is calculated as follows: J (t) = 8 ∆t2 [1 − | ⟨Ψ(t) | Ψ(t + ∆t)⟩ |], (F1) where ∆t is a sufficiently small increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The fidelity | ⟨Ψ(t) | Ψ(t + ∆t)⟩ | can be computed by the two-sided Lindblad equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (C2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The quantum dynamical activity is defined by [64] B(t) ≡ J (t) t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (F2) Appendix G: Linear response Here, we show detailed calculations of the linear re- sponse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us consider applying a weak pertur- bation χFf(t) to the master equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Considering the perturbation expansion with respect to χ, upto the first order, the probability distribution is expanded as P(t) = Pst + χP1(t), (G1) where P1(t) is the first-order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G1) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1), we have d dt (Pst + χP1(t)) = (W + χFf(t)) (Pst + χP1(t)) , (G2) in which collecting the terms with respect to the order of χ yields O(χ0) d dtPst = WPst, (G3) O(χ1) d dtP1(t) = WP1(t) + FPstf(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G4) The zeroth order equation vanishes in definition since Pst is assumed to be the stationary solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' P1(t) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (15) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In the main text, we consider a scoring function G(Bn) and its expecta- tion given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The change of ⟨G⟩ due to the perturbation can be represented by ∆G(t) ≡ χ1GP1(t) = χ � t −∞ 1GeW(t−t′)FPstf(t′)dt′ = χ � ∞ −∞ RG(t − t′)f(t′)dt′, (G5) 9 where RG(t) is the linear response function [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (18)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (3), the time derivative of C(t) reads d dtC(t) = 1SeWtWSPst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G6) In the main text, we consider the case G = S and F = WS, which will be assumed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The perturbation WS can be expressed by WS = � ���� S11W11 S22W12 · · SNNW1N S11W21 S22W22 SNNW2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' S11WN1 S22WN2 · · · SNNWNN � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G7) We immediately obtain RS(t) = d dtC(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G8) For the pulse perturbation, f(t) = δ(t), where δ(t) is the Dirac delta function, we obtain ∆S(p)(t) = χ � ∞ −∞ RS(t − t′)δ(t′)dt′ = χRS(t) = χ d dtC(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G9) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (7), we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Next, we consider the step perturbation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=', f(t) = Θ(t), where Θ(t) is the Heaviside step function: Θ(t) = � 0 (t < 0) 1 (t ≥ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (G10) Then we have ∆S(p)(t) = χ � ∞ −∞ RS(t − t′)Θ(t′)dt′ = χ � t 0 RS(t − t′)dt′ = χ � t 0 RS(t′)dt′ = χ � t 0 dC(t′) dt′ dt′ = χ (C(t) − C(0)) , (G11) which yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (21) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Appendix H: Norm and distance measures For readers’ convenience, we here review the norm and distance measures for quantum and classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let A and B be arbitrary Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The Shattan p-norm is defined by ∥A∥p ≡ � Tr ��√ A2 �p�� 1 p = � � � λ∈evals(A) |λ|p � � 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H1) For particular p, we have ∥A∥op = ∥A∥∞ = max λ∈evals(A) |λ|, (H2) ∥A∥tr = ∥A∥1 = Tr �√ A2 � , (H3) ∥A∥hs = ∥A∥2 = � Tr [A2], (H4) where evals(A) gives a set of eigenvalues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Equa- tions (H2), (H3), and (H4) are referred to as the opera- tor norm, the trace norm, and the Hilbert-Schmidt norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The H¨older inequality states |Tr [AB]| ≤ ∥A∥p ∥B∥q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H5) where p and q should satisfy 1/p+1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' When p = q = 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H5) reduces to the Cauchy-Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' In particular, we use p = ∞ and q = 1 case: |Tr [AB]| ≤ ∥A∥op ∥B∥tr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H6) Let us define the trace distance and quantum fidelity: TD(ρ, σ) ≡ 1 2 ∥ρ − σ∥1 , (H7) Fid(ρ, σ) ≡ � Tr �√ρσ√ρ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H8) When considering pure states |ψ⟩ and |φ⟩, the fidelity reduces to the inner product: Fid (|ψ⟩ , |φ⟩) = |⟨ψ|φ⟩|2 , (H9) TD(|ψ⟩ , |φ⟩) = � 1 − | ⟨ψ|φ⟩ |2 (H10) These two distances are related via TD(ρ, σ) ≤ � 1 − Fid(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H11) The equality of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H11) holds when both ρ and σ are pure [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let us introduce related classical probability distance measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' Let p(x) and q(x) be probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' The total variation distance and the Hellinger distance are given, respectively, by TVD(p, q) ≡ 1 2 � x |p(x) − q(x)|, (H12) Hel2(p, q) ≡ 1 2 � x �� p(x) − � q(x) �2 (H13) = 1 − Bhat(p, q) (H14) 10 where Bhat(p, q) is the Bhattacharyya coefficient: Bhat (p, q) ≡ � x � p(x)q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H15) Between the total variantion and the Hellinger distances, the following relations are known to hold [76]: Hel2(p, q) ≤ TVD(p, q) (H16) ≤ � Hel2(p, q)(2 − Hel2(p, q)) (H17) ≤ � 2Hel2(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQfSANm/content/2301.03060v1.pdf'} +page_content=' (H18) ACKNOWLEDGMENTS This work was supported by JSPS 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University + College Station, Texas, US + spgbarrett@tamu.edu +Yingjie Hu +Department of Geography + University at Buffalo + Buffalo, New York, US + yhu42@buffalo.edu +Lei Zou +Department of Geography + Texas A&M University + College Station, Texas, US + lzou@tamu.edu +Yi Qiang +School of Geoscience + University of South Florida +Tampa, Florida, US +qiangy@usf.edu +ABSTRACT +Extracting precise geographical information from textual contents +is crucial in a plethora of applications. For example, during +hazardous events, a robust and unbiased toponym extraction +framework can provide an avenue to tie the location concerned to +the topic discussed by news media posts and pinpoint humanitarian +help requests or damage reports from social media. Early studies +have leveraged rule-based, gazetteer-based, deep learning, and +hybrid approaches to address this problem. However, the +performance of existing tools is deficient in supporting operations +like emergency rescue, which relies on fine-grained, accurate +geographic information. The emerging pretrained language models +can better capture the underlying characteristics of text information, +including place names, offering a promising pathway to optimize +toponym recognition to underpin practical applications. In this +paper, TopoBERT, a toponym recognition module based on a one- +dimensional Convolutional Neural Network (CNN1D) and +Bidirectional Encoder Representation from Transformers (BERT), +is proposed and fine-tuned. Three datasets (CoNLL2003-Train, +Wikipedia3000, WNUT2017) are leveraged to tune the +hyperparameters, discover the best training strategy, and train the +model. Another two datasets (CoNLL2003-Test and Harvey2017) +are used to evaluate the performance. Three distinguished +classifiers, linear, multi-layer perceptron, and CNN1D, are +benchmarked to determine the optimal model architecture. +TopoBERT achieves state-of-the-art performance (f1-score=0.865) +compared to the other five baseline models and can be applied to +diverse toponym recognition tasks without additional training. +KEYWORDS +Natural Language Processing; Geoparser; Convolutional Neural +Network; Toponym Recognition; BERT + +1 Introduction +Since the emergence of social sensing, scholars have been +endeavoring to sense the pulse of society with the help of satellite +images, sensor networks from IoT and various forms of textual +information from the Internet. Extra attention has been paid to +mining knowledge from social media because people nowadays are +consciously or unconsciously sharing their views towards ongoing +events online, which propels social media to become one of the few +agents that reflects the real-time societal awareness, reactions and +impacts of particular events. This trait is a rare feature seldom +shared by other forms of data sources. +In the light of this feature, Avvenuti et al. presented an early +earthquake detecting and warning system using Twitter data, which +offers prompt detection of events [1]. Several case studies +processed social media data with geocoding and sentiment analysis +tools to analyze the spatial patterns of changing public awareness +and emotions toward hurricanes in different phases of the disaster +management cycle [2,3]. Huang et al. scrutinized the human +mobility patterns during the COVID-19 pandemic at multiple +scales based on geotagged Twitter data [4]. Zhou et al. proposed +VictimFinder which is capable of harvesting social media help +requests during hurricanes [5]. +Let alone the fact that geographical information being one of the +key elements of knowledge generation, the aforementioned studies +and other similar spatial analysis and modeling are highly +dependent on the location information of the social media data. +However, social media users start to pay more attention to user +privacy, which results in a significant drop of the number of +geotagged tweets. Simultaneously, Twitter published policies +forbidding users to attach precise longitudes and latitudes to tweets. +Moreover, the geographical information bound up with the social +media posts might not necessarily be equivalent to the place names +described in the textual content of the post. Thus, extracting +location information from the textual content of social media data +has inevitably become an issue that needs to be addressed. This +breeds the process of geoparsing, a two-step approach which +includes toponym recognition (identifying place names from texts) +and toponym resolution (transforming location names to +geographical coordinates). This paper focuses on the first +component of geoparsing. + + + + + +Existing studies on toponym recognition can be categorized into +four parties based on the character of the solutions, namely rule- +based, gazetteer-based, statistical learning-based, and hybrid +approaches. In general, statistical learning and hybrid methods that +incorporate deep learning techniques render better performance +than methods that solely rely on rules or gazetteers [6,7,8,9]. Based +on Bidirectional Long Short-Term Memory (BiLSTM), Wang et al. +introduced NeuroTPR to extract place names [6]. Qi et al. extended +CoreNLP and brought about an open-sourced named entity +recognition python toolkit called Stanza, which is able to detect +place names and support multiple languages [7]. SAVITR is a +system that combines both NLP techniques and gazetteers for real- +time location extraction [8]. Hu et al. addressed the incompleteness +of gazetteers and fused gazetteers, rules, and deep learning to +render a reliable place name extractor, GazPNE [9]. +However, those studies suffer from several limitations. First, some +models do not focus only on place names, so their prediction of +location name extraction might be disturbed. Second, recurrent +neural network based deep learning models might suffer from +information vanishing problems when the input sequence gets +larger and network deeper. Third, complicated deep neural +networks frequently require large, annotated datasets and are time- +consuming to train to achieve promising results. +To address the aforementioned latent flaws, this paper proposes +TopoBERT, a toponym recognition module based on a one- +dimensional +Convolutional +Neural +Network +(CNN) +and +Bidirectional Encoder Representation from Transformers (BERT). +It contributes in the following directions. First, several classifiers +were tested and one feasible model and classifier combination +based on the evaluation result of a standard dataset is determined. +Second, TopoBERT was tested by an unseen dataset together with +some other existing tools to verify its generalizability. Third, the +tool is ready-to-use and the dataset we generated in this study can +be used by other scholars to train, test, and compare different +toponym recognition models and tools. +The remainder of this paper is structured as follows. The datasets +involved in fine-tuning and testing the framework, a concise +introduction of the holistic design of the framework, the +implementation of the framework, and the parameters used in fine- +tuning the framework are detailed in section 2. The results of the +experiments conducted are documented in section 3. Section 4 +illustrates the potential limitations of this work and lists several +future research directions. Section 5 epitomizes the findings of this +paper and presents the implications of this study. +2 Methodology +2.1 Datasets +Totally four different datasets were utilized to train the module and +evaluate the performance. CoNLL2003 is a shared task that +concerns named entity recognition, which has been widely applied +to training deep learning models [10]. The data contains entities of +five types: persons (PER), organizations (ORG), locations (LOC) +and miscellaneous names (MISC) and other words that are +irrelevant to named entities of the aforementioned four groups (O). +The prefix “B-” and “I-” are used to tag the beginning of a named +entity and words that fall inside a named entity [10]. The dataset is +originally divided into training, validation, and test data which are +noted +as +CoNLL2003-Train, +CoNLL2003-Validation +and +CoNLL2003-Test. Training data is used to train a deep learning +model, validation data is used to tune the hyperparameters of the +model, and the test data is used to evaluate the performance of the +trained model. The data distribution of each label type in the three +datasets is depicted in Figures 1(a), 1(b), and 1(c), respectively. The +dataset is later modified to suit the purpose of this study by labeling +all the named entities as “O” except for the location entities. +Around 4.1% of the tags are location entities in these datasets. + + + +(a) (b) (c) +Figure 1: Data Distribution of CoNLL2003 Dataset +WNUT2017 is a relatively smaller dataset collected from Twitter +and manually annotated, the objective of which is to tackle the +issues caused by novel, emerging, singleton named entities in noisy +text [11]. It aims to offer support to sustainable named entity +recognition systems. This dataset contains seven different groups: +person, location, corporation, product, creative work, group and +none of the above. Considering the main focus of this paper and +different tags used to label the dataset, this dataset is preprocessed +to retain only the location entities tag and to unify the tag symbols +used based on CoNLL2003 (location entities are tagged with “B- +LOC” or “I-LOC” while the rest are tagged with “O”). The +distribution of data under each label type in the modified dataset is +shown in Figure 2(a). The total number of location names in this +dataset is 1140. + + + + +(a) + (b) (c) +Figure 2: Data Distribution of WNUT2017, Wiki300 and +Harvey2017 Dataset +Wiki3000 is an automatically generated dataset from Wikipedia +articles by a data producing workflow proposed by Wang et al. [6]. +The proposed auto-annotation approach utilizes the first paragraph +of Wikipedia articles which usually encompass various entities +presented with hyperlinks. These hyperlinks are later checked if +they are associated with a geographical location. If so, the + +CoNLL2003-TrainDataset +200000 +169578 +150000 +Count +100000 +50000 +82971002511128 +4593 +0 +LOCORGPER +RMISO +ClassNamesCoNLL2003-ValidationDataset +50000 +42759 +40000 +Count +30000 +20000 +10000 +3149 +20942092 +1268 +0 +LOCORG +PER +MISC +ClassNamesCoNLL2003-TestDataset +4000038323 +30000 +Count +20000 +10000 +192524962773 +918 +0 +LOCORG +PER +MISC +ClassNamesWNUT2017Dataset +10673 +10000 +Count +5000 +1140 +0 +0 +LOC +ClassNamesWiki3000Dataset +40466 +40000 +30000 +Count +20000 +16000 +10000 +0 +0 +LOC +ClassNamesHarvey2017Dataset +15295 +15000 +Count +10000 +5000 +3973 +0 +0 +LOC +ClassNames + + +hyperlinked word will be labeled as a toponym. Then the Wikipedia +article is divided into multiple short sentences within 280 +characters with additional strategies such as random flipping to +mimic the general patterns of Twitter posts [6]. The distribution of +data under each label type is shown in Figure 2(b). +Harvey2017 is a dataset originally collected from the North Texas +University repository (https://digital.library.unt.edu/ark:/67531 +/metadc993940/), which contains 7,041,866 tweets collected based +on hashtag query. It was pruned, randomly subsampled and +manually annotated by Wang et al. to form a new dataset with 1000 +tweets aiming to evaluate NeuroTPR [6]. This dataset is adopted by +this paper to test the performance of TopoBERT. The distribution +of data under each label type is shown in Figure 2(c). +2.2 Framework Design and Implementation +As mentioned in section 1, there is an acute conflict between robust +spatial analysis on social media or news media and the diminishing +availability of geolocated textual context. Additionally, the location +mentioned in the textual content of the tweets might differ from the +geotags attached. A reliable and ready-to-use geoparser can be the +mediator of such conflicts. Therefore, we present a general location +extractor that can be used upon social media and news media. The +workflow is shown in Figure 3. +The existing geotags of the data will be retained, and the textual +contents will go through a rule-based data preprocessing module +before they are fed to a zip code extractor and place name extractor. +Once the place names are pulled out, a geocoding service will be +applied to transform the place names into precise coordinates. The +place name extractor is marked with an orange dashed rectangle in +Figure 3 and serves as the crucial backbone of the entire workflow. + + +Figure 3: Holistic Design of Location Extraction Framework +for Textual Contents + +Figure 4: Demonstration of token classification workflow. +Identifying location names from input sentences is a token +classification task (Figure 4), which contains two parts. A language +model and a classifier. It behaves similar to how human beings +analyze whether the given words are place names or not. First the +language model attempts to understand the language by +transforming the tokenized input data into higher dimensional +space which captures the meaning of words in a given sentence, +then the classifier makes predictions based on the transformed +vectors and determines whether the input word belongs to location +entity. +The heart of the proposed toponym recognition module, +TopoBERT, is the Bidirectional Encoder Representation from +Transformers (BERT). It is structured by stacking the encoder +components of the Transformer architecture and is designed to be +pretrained in an unsupervised manner. BERT takes advantage of +the Attention [25] mechanism, which resolves the information +vanishing issue that often upsets recurrent neural networks such as +Long Short-Term Memory [26] and Gated Recurrent Neural +Network [27] when the input sequence gets longer. Moreover, +distinguished from many other bidirectional language models, such +as ELMo designed by Peters et al. [28], in which the contextual +representation of every word is the concatenation or summation of +the forward and backward representations, BERT reads the entire +sequence of words at once and is trained using a Masked Language +Model (MLM) approach and a Next Sentence Prediction (NSP) +approach which genuinely implemented the bidirectional concept +or unidirectional concept. These two features combined facilitate +better language understanding and bring the trophy to BERT + +Geotag +Geotagged +Coordinates +Bounding +Box +Center of Bounding +Box +Place +Social Media +Coordinatesfrom +Name +Data +PlaceName +No +4 +ZipCode +Rule-based +Extractor +Data +Google +ZipCodes +NewsMedia +Geocoding +Preprocess +Data +Toponym +IdentifiedLocation +Recognition +Names +Coordinatesfrom +GeocodingTokenClassification +B-LOC +I-LOC +Outputpredictionforeach +token +Classifier +Language Model +［101,1030,17870,...102,... +0, +0, +0] +Tokenizer +#HarveyRescueHoustonTX77074waitingforwater +rescueintheattic.Pleasehelp!" + + + +throughout a number of NLP tasks under the General Language +Understanding Evaluation (GLUE) benchmark [12]. +Off-the-shelf pretrained BERT model weights can be separated into +several categories based on the size of the model, whether upper +and lower cases are taken into consideration, the targeted language, +and +unique +training +strategies +(https://huggingface.co/transformers/v3.3.1/pretrained_models.ht +ml). Since place names are highly case sensitive and only the +English language is involved in this study, ‘bert-base-cased’ and +‘bert-large-cased’ are selected as the candidate pretrained models +to be evaluated. The ‘bert-base-cased’ model comprises 12 layers, +and each hidden layer has 768 nodes, with 12 self-attention heads +and a total number of 110 million parameters. The ‘bert-large-cased’ +model consists of 24 layers, and each hidden layer has 1024 nodes, +with 16 self-attention heads and 340 million parameters. The +parameters are pretrained with English text from BooksCorpus +(800 million words) and English Wikipedia (2,500 million words). +By stacking a classifier on top of BERT, the combo can be fine- +tuned to accomplish this downstream. Recent study showed that +model performance can be enhanced by applying classifiers more +complex than simple linear classifier or Conditional Random Field +(Zhou et al. 2022). Therefore, three classifiers were examined in +this study, namely linear classifier, multi-layer perceptron (MLP, +Figure 5) and one-dimensional CNN (CNN1D, Figure 6). The +simple linear classifier connects the output of the language model +to the final prediction results with the softmax activation function. +MLP applied in this study contains three fully connected layers and +links the language model output with a layer with the input size +equivalent to the output vector size. The number of hidden layer +nodes is 256 and the output layer size equals the number of distinct +labels from the training dataset. The CNN models are competent in +detecting underlying features [29] and one-dimensional CNN has +been successfully applied to process natural language [30, 31]. +Realizing +location +names +might +share +some +common +characteristics, the idea of CNN1D is adopted. The vector output of +the language model can be considered as a one-dimensional signal +and a CNN1D with kernel size 3 is applied. The output channel of +the convolution is 16. Followed by a max pooling layer of size 2, +which further generalizes the features and reduces model +complexity. All channels of the max pooling layer output are +concatenated into a single vector and is fed to a fully connected +MLP with hidden layer size equals to 128. +All model combinations were implemented using Python language +and pertinent packages. The dataset splitting took advantage of the +ScikitLearn library and the BERT models were implemented based +on +the +huggingface +Transformer +library +(https://huggingface.co/transformers/). The model finetuning +pipeline was built using PyTorch functions. + + + + +Figure +5: +TopoBERT +Architecture +with +Multi-layer +Perceptron as Classifier + +Figure 6: TopoBERT Architecture with One-Dimensional +Convolutional Neural Network as Classifier +2.3 Training and Evaluation +TopoBERT is envisioned to be a ready-to-use module that renders +optimal performance in toponym recognition. Models with +different architectures were trained and evaluated with six datasets +specified in Section 2.1 to determine the best model architecture +and training strategy. The training process utilized CoNLL2003- +Train as the training dataset by default and compared to another +larger dataset fusing CoNLL2003, Wiki3000, and WNUT2017. +The original dataset is labelled at word-level which cannot be input +to BERT directly due to BERT’s word-piece encoding, otherwise +it will lead to large numbers of out of vocabulary words. To tackle + +BERT +E(cLs +En +ESEP +StackedTransformerEncoders +食 +价 +EICLS) +E +Em +ER +EISER +[CLS] +TOKEN 1 +TOKEN.m +TOKENn +[SEP] +Word +Embeddings +Multi-layer +Perceptron +"TheEuropeanComissionsaidonTuesday...tohumanbeings."Max +Pooling +Word +Embeddings +Multi-layer +Convolutional +Concatenated +Perceptron +Layers +Layers +BERT +ECLs +Ei +En +ESEP)] +StackedTransformerEncoders +食 +E(cLs) +E +E +En +E(SEP) +[CLS] +TOKEN1 +TOKEN +TOKENT +[SEP] +"The European Comission saidon Tuesday...tohuman beings." + + +with this issue, we first split the input data at word-level, and +applied BERT word-piece tokenizer to each word. The same label +was assigned to each word-piece of a single word. The labeled +word-pieces are then merged to form the new input data which +could be processed by BERT. This experiment aimed at measuring +the performance fluctuations caused by training data size and +heterogeneity. CoNLL2003-Validation was used during the +training process to tune several fundamental hyperparameters such +as training epochs and learning rate. CoNLL2003-Test and +Harvey2017 datasets were used to evaluate the model performance. +The Harvey2017 dataset was also used to benchmark TopoBERT +with five prevailing toponym recognition models, namely Stanford +NLP [32], spaCy (https://spacy.io/), Bidirectional LSTM-CRF [33], +DM_NLP [34], and NeuroTPR [6]. +The parameters of the classifier component of the module were +initialized with random non-zero numbers and the BERT +component was initialized with pre-trained parameters. The entire +module was trained with the fine-tuning approach [12], and the +parameters were updated using a mini-batch gradient descent +approach with early stopping. The maximum length of the input +sequence was limited to 128 in this paper. The maximum number +of training epochs was set to 50. As recommended by the original +BERT paper, the initial learning rate and the training batch size +were set to 2e-5 and 32 respectively [12]. Most commonly used loss +function for multi-class classification task, the cross-entropy loss +was employed. AdamW was selected as the optimizer during +training which adjusts the learning rate dynamically to accelerate +parameter convergence and implements weight decay to lower the +chance of overfitting. Warm up steps, which is using a very low +learning rate for the first several weight updating iterations, were +also introduced during training to reduce the impact of deviating +the model drastically from sudden exposure to unseen datasets. +Three commonly used evaluation metrics, precision, recall, and F1- +score (Equation 1-3), were applied to gauge the performance and +bias of the models. Precision calculates the percentage of correctly +identified location names (noted as True Positives, TP) among all +the location names predicted by the model, which combines both +TP and False Positives (FP). Recall measures the percentage of +correctly identified ones amongst all ground truth, which is the +combination of TP and False Negatives (FN). F1-score is the +harmonic mean of precision and recall, providing a comprehensive +metric to evaluate model performance. + 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = +�� +����� (Equation 1) + 𝑅𝑒𝑐𝑎𝑙𝑙 = +�� +����� + (Equation 2) +𝐹1–𝑠𝑐𝑜𝑟𝑒 = 2 ∗ +���������∗ ������ +���������� ������ + (Equation 3) + +The outputs of BERT models are at word-piece level and they are +concatenated using the special prefix ‘##’ and the word-level labels +are assigned base on the starting word-piece of the word. The +evaluation metrics are based on ‘per-token’ scores. Additionally, +location name entity consists of two types of labels (B-LOC and I- +LOC). In order to gauge the comprehensive performance of the +model on toponym recognition, the evaluation metrics were +calculated using a micro average approach, which computes a +global average of precision, recall, and F1-score. It calculates the +TP, FP and FN by counting the total number of TP, FP and FN +under each class, namely, “B-LOC” and “I-LOC”. + +3 Results and Analysis +The first step of the experiment targeted at determining the optimal +pretrained parameters for BERT model. We hypothesize that larger +models outperform smaller models. To verify this hypothesis, the +performance of the models initialized with ‘bert-base-cased’ and +‘bert-large-cased’ with a linear classifier stacked on top were tested. +The results are displayed in Table 1. +Table 1: Evaluation results for testing on different pretrained +parameters. +BERT Model +Classifie +r +Precisio +n +Recal +l +F1- +score +bert-base-cased +Linear +0.900 +0.904 +0.902 +bert-large- +cased +Linear +0.934 +0.901 +0.917 + +These two models were trained with CoNLL2003-Train and +evaluated with CoNLL2003-Test. Compared to ‘bert-base-cased’, +the precision of the prediction increased from 0.900 to 0.934 by +using ‘bert-large-cased’ while the recall almost remained static. +The F1-scores showed that ‘bert-large-cased’ rendered better +results which is in conformity with the original BERT paper [12] +and validated our initial hypothesis. Therefore, ‘bert-large-cased’ +was harnessed in all the follow-up experiments. +The second step of the experiments aimed to measure the influence +of the training data and determine the optimal classifier. The model +performances were evaluated using two different datasets, +CoNLL2003-Test and Harvey2017. We hypothesize that (a) the +model with CNN1D classifier yield better results and (b) models +trained with larger datasets perform better in placename recognition. +Table 2 and Table 3 list the evaluation metrics of all the tests. +Table 2: Evaluation results with CoNLL2003-Test dataset for +testing on training data variation and classifier types. +Training Data +Classifier +Precision +Recall +F1-score +CoNLL2003 +Linear +0.934 +0.901 +0.917 +CoNLL2003 +MLP +0.904 +0.910 +0.907 +CoNLL2003 +CNN1D +0.923 +0.920 +0.921 +Combined +Linear +0.889 +0.844 +0.866 +Combined +MLP +0.941 +0.884 +0.912 +Combined +CNN1D +0.942 +0.916 +0.929 +Table 3: Evaluation results with Harvey2017 dataset for +testing on training data variation and classifier types. + + + + + +Training Data +Classifier +Precision +Recall +F1-score +CoNLL2003 +Linear +0.895 +0.804 +0.847 +CoNLL2003 +MLP +0.885 +0.811 +0.846 +CoNLL2003 +CNN1D +0.898 +0.835 +0.865 +Combined +Linear +0.872 +0.589 +0.703 +Combined +MLP +0.932 +0.541 +0.685 +Combined +CNN1D +0.941 +0.668 +0.781 +The “CoNLL2003” under the Training Data column means +CoNLL2003-Train dataset and the “Combined” represents the +dataset merging CoNLL2003-Test, Wiki3000 and WNUT2017. +In Table 2, when models were trained with CoNLL2003-Train, the +one with a simple linear classifier produced the best precision +(0.934), and the one with CNN1D produced the best recall (0.920) +and F1-score (0.921). MLP performed the worst among the three +classifiers. When models were trained with a combined dataset, the +model with CNN1D outperformed the rest in all three metrics with +precision equal to 0.942, recall of 0.916, and F1-score of 0.929. The +one with a linear classifier produced the worst results with an F1- +score of 0.866. In Table 3, when models were trained with +CoNLL2003-Train, the one with the CNN1D classifier +outperformed the rest with precision equal to 0.898, recall of 0.835, +and F1-score of 0.865. When models were trained with a combined +dataset, the model with CNN1D successfully defended its trophy +by rendering precision of 0.941, recall of 0.668, and F1-score of +0.781. The models with MLP worked slightly worse than the ones +with linear classifiers. + The above elucidation certifies the hypothesis that models with +CNN1D generate the optimal performance. It also shows that more +complicated classifiers like multi-layer perceptron do not +necessarily render better results. +However, when viewing Tables 2 and 3 contemporaneously, the +results from training with different datasets, the metrics indicated +that the model trained with the combined dataset generally +performed worse than the ones trained with merely CoNLL2003- +Train. This phenomenon contradicts the hypothesis that models +trained with larger datasets perform better. After scrutinizing the +dataset used for training, we noticed some inconsistencies in the +labeling criteria of the datasets. Some examples are listed in Table +4 and the unexpected phenomenon can be interpreted by the +heterogeneity of the datasets. +Table 4: Examples of different labels across the datasets used +for training the model. +Example Entity +Dataset +CoNLL200 +3 +Wiki300 +0 +WNUT201 +7 +"Canadian" +B-MISC +O +B-LOC +"Planet" +O +O +B-LOC +"east" +O +O +B-LOC +"orchard" +"academy" +B-ORG/ +I-ORG +O +B-LOC/ +I-LOC +"earth" +O +N/A +B-LOC +It can be seen from Table 4 that the word “Canadian,” which is +labeled as “B-MISC” (beginning of a miscellaneous name), is +identified as “B-LOC” (beginning of a location) in the WNUT2017 +dataset. The words “Planet”, “east,” and “earth” are misclassified +as locations in the WNUT2017 dataset. The phrase “orchard +academy,” regarded as an organization under the CoNLL2003 +criteria, is also labeled as a location entity. In this case, combining +several heterogeneous datasets can be considered adding some +helpful unseen samples to the original training data while +introducing a substantial amount of noise. +Rolnick et al. [13] experimented on several deep learning models +when trained with noisy data and claimed that the CNN model is +more resilient to noise than MLP and linear models. The trend of +performance change shown in Tables 2 and 3 when trained with +different datasets is in accordance with this statement. It is +noticeable that the models experience an increase in precision and +a drastic decrease in recall when trained with a combined dataset. +This incident can as well be triggered by noisy data. Since deep +learning models attempt to learn the underlying patterns of the +training data, the existing noise will confuse the model, resulting in +a fewer number of positive predictions. This might result in an +increase in precision and a decrease in recall. +Based on the observation and interpretation above, the BERT +model initialized with ‘bert-large-cased’, stacked with a CNN1D +classifier and fine-tuned with CoNLL2003-Train was selected as +the finalized TopoBERT module. Table 5 shows a comparison +between TopoBERT and five other models and tools based on the +Harvey2017 dataset. +Table 5: Evaluation results with Harvey2017 dataset for +comparing TopoBERT with other existing models. +Model +Precisio +n +Recal +l +F1- +score +Stanford NER (broad +location) +0.729 +0.440 +0.548 +SpaCy NER (broad location) +0.461 +0.304 +0.366 +BiLSTM-CRF +0.703 +0.600 +0.649 +DM_NLP +0.729 +0.680 +0.703 +NeuroTPR +0.787 +0.678 +0.728 +TopoBERT +0.898 +0.835 +0.865 +The SpaCy version v3.0 is used with model “en_core_web_sm” +loaded. Broad location indicates that we include entities in both +LOCATION and ORGANIZATION for Stanford NER, and we +include entities in the types of LOC, ORG, FACILITY, and GPE +for spaCy NER. Evaluation results show that TopoBERT prevailed +in the competition with precision equals to 0.898, recall 0.835 and +F1-score 0.865. This result outperformed other baseline models by +at least 18%. +TopoBERT has been developed as a ready-to-use module. The +output data of TopoBERT includes word labels and confidence of +the prediction. It complies with JSON file format for ease of use. + + + + +The source code has been uploaded to GitHub and can be accessed +with the link: https://github.com/SPGBarrett/gearlab_topobert. + +4 Discussion +This paper presents a geoparsing framework and breeds a plug and +play toponym recognition module which can facilitate spatial +analysis based on social media or news media data. Figure 7 shows +a practical application of this framework in locating Twitter posts +under fine-grained topics during hazardous events. The study area +is the State of Florida, and the dots in multiple colors displayed on +the map are tweets posted during Hurricane Irma harvested by +Twitter developer API. The locations of those tweets without +geotags are retrieved by running TopoBERT and google geocoding +service. The module also enjoys the potential of being used for +location name detection for news media to pinpoint the discussed +topics [14,15] and help to identify fake news [16]. + + +Figure 7: Toponym recognition applied to locate Twitter posts +during disasters. +This paper concentrates mainly on designing a novel architecture +of a reliable and versatile module for toponym recognition. +However, the performance enhancement can continue by +addressing the following issues. +First, the models are trained and evaluated based on well prepared +datasets. This can be regarded as a best-case scenario compared to +real life situations. Place name usage can be highly ambiguous and +random, especially within social media platforms. Typos are +extremely common which might cause out-of-vocabulary words in +language models. Place name abbreviations such as “Boulevard” +and “blvd”, “Drive” and “Dr.”, “Street” and “St.” and so forth are +frequently utilized interchangeably. People might unconsciously +ignore the correct upper-case and lower-case usage, such as +“college station” and “College Station”, “mexico” and “MEXICO”. +Meticulous data preprocessing methods can be incorporated to +tackle this problem in order to achieve better overall performance. +Second, several rule-base approaches can be leveraged to further +boost the performance. Enlightened by the success of hybrid +models [9], sets of grammar rules based on the composition of +nouns, determiners, adjectives, conjunctions, numbers and +possessive ending can be designed [17]. Additionally, commonly +used gazetteers such as OpenStreetMap and GeoNames can be used +as extra named entity matching criteria which will enhance the True +Positives of the model. Regional criteria can be appended to the +model while identifying place names by making country name, +state names, county names, or bounding boxes as input variables of +the model. This will allow the model to add constraints during +processing. The top-N words from word embedding models [9,35], +which are not place names, can be applied to filter words during +data preprocessing. This will to some extent eliminate the False +Positives of the prediction. +Third, due to the data-hungry nature of deep learning, data +availability and quality are topics being inevitably discussed when +large complicated deep learning models are involved. It is common +knowledge in the deep learning world that larger datasets lead to +better generalizability and performance. However, this statement +fails to hold true in this paper due to the fact that the larger datasets +are derived from several distinguished smaller datasets labeled +under their own unique regime. Therefore, there is an urgent need +to define criteria and build unified datasets for toponym recognition +model training, evaluating and benchmarking. The dataset can be +manually modified based on existing datasets and augmented using +rule-based methods, gazetteers or Generative Adversarial Network +[18,19,20]. +Fourth, fine-tuned language models can be few-shot or zero-shot +learners, which means that the models can be applied directly to +certain downstream tasks with very little or even no further training +[21,22,23]. This is because advanced language models can better +capture the meaning of the text. This claim is also underpinned by +the result of this paper which leverages BERT to boost the module +capability. Therefore, incorporating gigantic models such as GPT- +3 [24] might lead to another round of performance enhancement. +5 Conclusion +To further enhance the performance of toponym recognition by +better understanding natural language, TopoBERT, which +incorporate pretrained language model, BERT, is introduced. +Experiments on the pretrained parameters, training dataset +combinations, and model architecture reveal the following findings. +First, the toponym recognition model performance is sensitive to +the architecture of pre-trained language models and classifiers. The +models initialized with a larger-structured BERT model (“bert- +large-cased”) show an advantage over the models initialized with a +basic BERT model (“bert-base-cased”). More complicated +classifiers like MLP do not necessarily win over simple linear +classifiers. Second, increasing training data size produces worse +results, especially for the recall, due to data heterogeneity. The +model trained with single dataset, CoNLL2003-Train, and stacked +on top with a CNN1D classifier renders the optimum results both +on CoNLL2003-Test and Harvey2017 datasets. Finally, the +developed TopoBERT module outperforms existing models in + +Hurricane Category +IrmaRouteLine +Florida_Census_ Tract_2019 +0 +Human_Help_Florida +Animal HelpFlorida +3 +Infrastructure Florida +ShelterFlorida + + + +recognizing place names in texts. The clinched TopoBERT with the +optimal model architecture and training strategy produces reliable +toponym prediction and achieves F1-score of 0.865 on Harvey2017 +dataset, which surpasses other prevailing models or tools by at least +18%. +In nutshell, the discoveries of this paper contribute in determining +the optimal model structure on toponym recognition tasks and +urges a large standardized dataset labeled with unified regime to +support model training and benchmarking. A plug and play module +is implemented and open sourced to support pertinent applications +and similar research. +ACKNOWLEDGMENTS +The research is supported by a project funded by the U.S. National +Science Foundation: Reducing the Human Impacts of Flash Floods +- Development of Microdata and Causal Model to Inform +Mitigation and Preparedness (Award No. 1931301). +REFERENCES +[1] Marco Avvenuti, Stefano Cresci, Mariantonietta La N. Polla, Andrea Marchetti, +and Maurizio Tesconi. 2014. Earthquake emergency management by Social +Sensing. 24 - 28 March 2014, Budapest, Hungary. IEEE, Piscataway, NJ. +[2] Lei Zou, Nina S. N. Lam, Shayan Shams, Heng Cai, Michelle A. Meyer, Seungwon +Yang, Kisung Lee, Seung-Jong Park, and Margaret A. Reams. 2019. Social and +geographical disparities in Twitter use during +[3] Lei Zou, Nina S. N. Lam, Heng Cai, and Yi Qiang. 2018. Mining Twitter Data +for Improved Understanding of Disaster Resilience. 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Bidirectional LSTM-CRF Models +for Sequence Tagging. + + + + +[34] Chunping Ma, Huafei Zheng, Pengjun Xie, Chen Li, Linlin Li, and Luo Si. +DM_NLP at SemEval-2018 Task 8: neural sequence labeling with linguistic +features. In Proceedings of The 12th International Workshop on Semantic +Evaluation. Association for Computational Linguistics, Stroudsburg, PA, USA, +707–711. DOI: https://doi.org/10.18653/v1/S18-1114. +[35] Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S. Corrado, and Jeff Dean. +Distributed Representations of Words and Phrases and their Compositionality. + + + diff --git a/A9FRT4oBgHgl3EQfujgS/content/tmp_files/load_file.txt b/A9FRT4oBgHgl3EQfujgS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7455fe036d98b9cfbd769d38010e022135e4134e --- /dev/null +++ b/A9FRT4oBgHgl3EQfujgS/content/tmp_files/load_file.txt @@ -0,0 +1,736 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf,len=735 +page_content='TopoBERT: Plug and Play Toponym Recognition Module Harnessing Fine-tuned BERT∗ Bing Zhou Department of Geography Texas A&M University College Station, Texas, US spgbarrett@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu Yingjie Hu Department of Geography University at Buffalo Buffalo, New York, US yhu42@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu Lei Zou Department of Geography Texas A&M University College Station, Texas, US lzou@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu Yi Qiang School of Geoscience University of South Florida Tampa, Florida, US qiangy@usf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu ABSTRACT Extracting precise geographical information from textual contents is crucial in a plethora of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' For example, during hazardous events, a robust and unbiased toponym extraction framework can provide an avenue to tie the location concerned to the topic discussed by news media posts and pinpoint humanitarian help requests or damage reports from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Early studies have leveraged rule-based, gazetteer-based, deep learning, and hybrid approaches to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, the performance of existing tools is deficient in supporting operations like emergency rescue, which relies on fine-grained, accurate geographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The emerging pretrained language models can better capture the underlying characteristics of text information, including place names, offering a promising pathway to optimize toponym recognition to underpin practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In this paper, TopoBERT, a toponym recognition module based on a one- dimensional Convolutional Neural Network (CNN1D) and Bidirectional Encoder Representation from Transformers (BERT), is proposed and fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Three datasets (CoNLL2003-Train, Wikipedia3000, WNUT2017) are leveraged to tune the hyperparameters, discover the best training strategy, and train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Another two datasets (CoNLL2003-Test and Harvey2017) are used to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Three distinguished classifiers, linear, multi-layer perceptron, and CNN1D, are benchmarked to determine the optimal model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' TopoBERT achieves state-of-the-art performance (f1-score=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865) compared to the other five baseline models and can be applied to diverse toponym recognition tasks without additional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' KEYWORDS Natural Language Processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Geoparser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Convolutional Neural Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Toponym Recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' BERT 1 Introduction Since the emergence of social sensing, scholars have been endeavoring to sense the pulse of society with the help of satellite images, sensor networks from IoT and various forms of textual information from the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Extra attention has been paid to mining knowledge from social media because people nowadays are consciously or unconsciously sharing their views towards ongoing events online, which propels social media to become one of the few agents that reflects the real-time societal awareness, reactions and impacts of particular events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This trait is a rare feature seldom shared by other forms of data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In the light of this feature, Avvenuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' presented an early earthquake detecting and warning system using Twitter data, which offers prompt detection of events [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Several case studies processed social media data with geocoding and sentiment analysis tools to analyze the spatial patterns of changing public awareness and emotions toward hurricanes in different phases of the disaster management cycle [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' scrutinized the human mobility patterns during the COVID-19 pandemic at multiple scales based on geotagged Twitter data [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' proposed VictimFinder which is capable of harvesting social media help requests during hurricanes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Let alone the fact that geographical information being one of the key elements of knowledge generation, the aforementioned studies and other similar spatial analysis and modeling are highly dependent on the location information of the social media data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, social media users start to pay more attention to user privacy, which results in a significant drop of the number of geotagged tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Simultaneously, Twitter published policies forbidding users to attach precise longitudes and latitudes to tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Moreover, the geographical information bound up with the social media posts might not necessarily be equivalent to the place names described in the textual content of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Thus, extracting location information from the textual content of social media data has inevitably become an issue that needs to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This breeds the process of geoparsing, a two-step approach which includes toponym recognition (identifying place names from texts) and toponym resolution (transforming location names to geographical coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This paper focuses on the first component of geoparsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Existing studies on toponym recognition can be categorized into four parties based on the character of the solutions, namely rule- based, gazetteer-based, statistical learning-based, and hybrid approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In general, statistical learning and hybrid methods that incorporate deep learning techniques render better performance than methods that solely rely on rules or gazetteers [6,7,8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Based on Bidirectional Long Short-Term Memory (BiLSTM), Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' introduced NeuroTPR to extract place names [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' extended CoreNLP and brought about an open-sourced named entity recognition python toolkit called Stanza, which is able to detect place names and support multiple languages [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' SAVITR is a system that combines both NLP techniques and gazetteers for real- time location extraction [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' addressed the incompleteness of gazetteers and fused gazetteers, rules, and deep learning to render a reliable place name extractor, GazPNE [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, those studies suffer from several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' First, some models do not focus only on place names, so their prediction of location name extraction might be disturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Second, recurrent neural network based deep learning models might suffer from information vanishing problems when the input sequence gets larger and network deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Third, complicated deep neural networks frequently require large, annotated datasets and are time- consuming to train to achieve promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' To address the aforementioned latent flaws, this paper proposes TopoBERT, a toponym recognition module based on a one- dimensional Convolutional Neural Network (CNN) and Bidirectional Encoder Representation from Transformers (BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It contributes in the following directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' First, several classifiers were tested and one feasible model and classifier combination based on the evaluation result of a standard dataset is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Second, TopoBERT was tested by an unseen dataset together with some other existing tools to verify its generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Third, the tool is ready-to-use and the dataset we generated in this study can be used by other scholars to train, test, and compare different toponym recognition models and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The datasets involved in fine-tuning and testing the framework, a concise introduction of the holistic design of the framework, the implementation of the framework, and the parameters used in fine- tuning the framework are detailed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The results of the experiments conducted are documented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Section 4 illustrates the potential limitations of this work and lists several future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Section 5 epitomizes the findings of this paper and presents the implications of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1 Datasets Totally four different datasets were utilized to train the module and evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' CoNLL2003 is a shared task that concerns named entity recognition, which has been widely applied to training deep learning models [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The data contains entities of five types: persons (PER), organizations (ORG), locations (LOC) and miscellaneous names (MISC) and other words that are irrelevant to named entities of the aforementioned four groups (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The prefix “B-” and “I-” are used to tag the beginning of a named entity and words that fall inside a named entity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The dataset is originally divided into training, validation, and test data which are noted as CoNLL2003-Train, CoNLL2003-Validation and CoNLL2003-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Training data is used to train a deep learning model, validation data is used to tune the hyperparameters of the model, and the test data is used to evaluate the performance of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The data distribution of each label type in the three datasets is depicted in Figures 1(a), 1(b), and 1(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The dataset is later modified to suit the purpose of this study by labeling all the named entities as “O” except for the location entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1% of the tags are location entities in these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' (a) (b) (c) Figure 1: Data Distribution of CoNLL2003 Dataset WNUT2017 is a relatively smaller dataset collected from Twitter and manually annotated, the objective of which is to tackle the issues caused by novel, emerging, singleton named entities in noisy text [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It aims to offer support to sustainable named entity recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This dataset contains seven different groups: person, location, corporation, product, creative work, group and none of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Considering the main focus of this paper and different tags used to label the dataset, this dataset is preprocessed to retain only the location entities tag and to unify the tag symbols used based on CoNLL2003 (location entities are tagged with “B- LOC” or “I-LOC” while the rest are tagged with “O”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The distribution of data under each label type in the modified dataset is shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The total number of location names in this dataset is 1140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' (a) (b) (c) Figure 2: Data Distribution of WNUT2017, Wiki300 and Harvey2017 Dataset Wiki3000 is an automatically generated dataset from Wikipedia articles by a data producing workflow proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The proposed auto-annotation approach utilizes the first paragraph of Wikipedia articles which usually encompass various entities presented with hyperlinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' These hyperlinks are later checked if they are associated with a geographical location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' If so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' the ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='hyperlinked word will be labeled as a toponym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Then the Wikipedia article is divided into multiple short sentences within 280 characters with additional strategies such as random flipping to mimic the general patterns of Twitter posts [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The distribution of data under each label type is shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Harvey2017 is a dataset originally collected from the North Texas University repository (https://digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu/ark:/67531 /metadc993940/), which contains 7,041,866 tweets collected based on hashtag query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It was pruned, randomly subsampled and manually annotated by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' to form a new dataset with 1000 tweets aiming to evaluate NeuroTPR [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This dataset is adopted by this paper to test the performance of TopoBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The distribution of data under each label type is shown in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='2 Framework Design and Implementation As mentioned in section 1, there is an acute conflict between robust spatial analysis on social media or news media and the diminishing availability of geolocated textual context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Additionally, the location mentioned in the textual content of the tweets might differ from the geotags attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' A reliable and ready-to-use geoparser can be the mediator of such conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Therefore, we present a general location extractor that can be used upon social media and news media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The workflow is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The existing geotags of the data will be retained, and the textual contents will go through a rule-based data preprocessing module before they are fed to a zip code extractor and place name extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Once the place names are pulled out, a geocoding service will be applied to transform the place names into precise coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The place name extractor is marked with an orange dashed rectangle in Figure 3 and serves as the crucial backbone of the entire workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Figure 3: Holistic Design of Location Extraction Framework for Textual Contents Figure 4: Demonstration of token classification workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Identifying location names from input sentences is a token classification task (Figure 4), which contains two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' A language model and a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It behaves similar to how human beings analyze whether the given words are place names or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' First the language model attempts to understand the language by transforming the tokenized input data into higher dimensional space which captures the meaning of words in a given sentence, then the classifier makes predictions based on the transformed vectors and determines whether the input word belongs to location entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The heart of the proposed toponym recognition module, TopoBERT, is the Bidirectional Encoder Representation from Transformers (BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It is structured by stacking the encoder components of the Transformer architecture and is designed to be pretrained in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' BERT takes advantage of the Attention [25] mechanism, which resolves the information vanishing issue that often upsets recurrent neural networks such as Long Short-Term Memory [26] and Gated Recurrent Neural Network [27] when the input sequence gets longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Moreover, distinguished from many other bidirectional language models, such as ELMo designed by Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [28], in which the contextual representation of every word is the concatenation or summation of the forward and backward representations, BERT reads the entire sequence of words at once and is trained using a Masked Language Model (MLM) approach and a Next Sentence Prediction (NSP) approach which genuinely implemented the bidirectional concept or unidirectional concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' These two features combined facilitate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='better language understanding and bring the trophy to BERT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Geotag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Geotagged ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Bounding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Center of Bounding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Social Media ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Coordinatesfrom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='PlaceName ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='ZipCode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Rule-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Extractor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Google ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='ZipCodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='NewsMedia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Geocoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Preprocess ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Toponym ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='IdentifiedLocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Recognition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Names ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Coordinatesfrom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='GeocodingTokenClassification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='B-LOC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='I-LOC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Outputpredictionforeach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Language Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='［101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='17870,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='102,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 0, 0, 0] Tokenizer #HarveyRescueHoustonTX77074waitingforwater rescueintheattic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='Pleasehelp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='" throughout a number of NLP tasks under the General Language Understanding Evaluation (GLUE) benchmark [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Off-the-shelf pretrained BERT model weights can be separated into several categories based on the size of the model, whether upper and lower cases are taken into consideration, the targeted language, and unique training strategies (https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='co/transformers/v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1/pretrained_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='ht ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Since place names are highly case sensitive and only the English language is involved in this study, ‘bert-base-cased’ and ‘bert-large-cased’ are selected as the candidate pretrained models to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The ‘bert-base-cased’ model comprises 12 layers, and each hidden layer has 768 nodes, with 12 self-attention heads and a total number of 110 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The ‘bert-large-cased’ model consists of 24 layers, and each hidden layer has 1024 nodes, with 16 self-attention heads and 340 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The parameters are pretrained with English text from BooksCorpus (800 million words) and English Wikipedia (2,500 million words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' By stacking a classifier on top of BERT, the combo can be fine- tuned to accomplish this downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Recent study showed that model performance can be enhanced by applying classifiers more complex than simple linear classifier or Conditional Random Field (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Therefore, three classifiers were examined in this study, namely linear classifier, multi-layer perceptron (MLP, Figure 5) and one-dimensional CNN (CNN1D, Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The simple linear classifier connects the output of the language model to the final prediction results with the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' MLP applied in this study contains three fully connected layers and links the language model output with a layer with the input size equivalent to the output vector size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The number of hidden layer nodes is 256 and the output layer size equals the number of distinct labels from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The CNN models are competent in detecting underlying features [29] and one-dimensional CNN has been successfully applied to process natural language [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Realizing location names might share some common characteristics, the idea of CNN1D is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The vector output of the language model can be considered as a one-dimensional signal and a CNN1D with kernel size 3 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The output channel of the convolution is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Followed by a max pooling layer of size 2, which further generalizes the features and reduces model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' All channels of the max pooling layer output are concatenated into a single vector and is fed to a fully connected MLP with hidden layer size equals to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' All model combinations were implemented using Python language and pertinent packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The dataset splitting took advantage of the ScikitLearn library and the BERT models were implemented based on the huggingface Transformer library (https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='co/transformers/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The model finetuning pipeline was built using PyTorch functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Figure 5: TopoBERT Architecture with Multi-layer Perceptron as Classifier Figure 6: TopoBERT Architecture with One-Dimensional Convolutional Neural Network as Classifier 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='3 Training and Evaluation TopoBERT is envisioned to be a ready-to-use module that renders optimal performance in toponym recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Models with different architectures were trained and evaluated with six datasets specified in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1 to determine the best model architecture and training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The training process utilized CoNLL2003- Train as the training dataset by default and compared to another larger dataset fusing CoNLL2003, Wiki3000, and WNUT2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The original dataset is labelled at word-level which cannot be input to BERT directly due to BERT’s word-piece encoding, otherwise it will lead to large numbers of out of vocabulary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' To tackle BERT E(cLs En ESEP StackedTransformerEncoders 食 价 EICLS) E Em ER EISER [CLS] TOKEN 1 TOKEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='m TOKENn [SEP] Word Embeddings Multi-layer Perceptron "TheEuropeanComissionsaidonTuesday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='tohumanbeings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' "Max Pooling Word Embeddings Multi-layer Convolutional Concatenated Perceptron Layers Layers BERT ECLs Ei En ESEP)] StackedTransformerEncoders 食 E(cLs) E E En E(SEP) [CLS] TOKEN1 TOKEN TOKENT [SEP] "The European Comission saidon Tuesday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='tohuman beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='" with this issue, we first split the input data at word-level, and applied BERT word-piece tokenizer to each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The same label was assigned to each word-piece of a single word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The labeled word-pieces are then merged to form the new input data which could be processed by BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This experiment aimed at measuring the performance fluctuations caused by training data size and heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' CoNLL2003-Validation was used during the training process to tune several fundamental hyperparameters such as training epochs and learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' CoNLL2003-Test and Harvey2017 datasets were used to evaluate the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The Harvey2017 dataset was also used to benchmark TopoBERT with five prevailing toponym recognition models, namely Stanford NLP [32], spaCy (https://spacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='io/), Bidirectional LSTM-CRF [33], DM_NLP [34], and NeuroTPR [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The parameters of the classifier component of the module were initialized with random non-zero numbers and the BERT component was initialized with pre-trained parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The entire module was trained with the fine-tuning approach [12], and the parameters were updated using a mini-batch gradient descent approach with early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The maximum length of the input sequence was limited to 128 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The maximum number of training epochs was set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' As recommended by the original BERT paper, the initial learning rate and the training batch size were set to 2e-5 and 32 respectively [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Most commonly used loss function for multi-class classification task, the cross-entropy loss was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' AdamW was selected as the optimizer during training which adjusts the learning rate dynamically to accelerate parameter convergence and implements weight decay to lower the chance of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Warm up steps, which is using a very low learning rate for the first several weight updating iterations, were also introduced during training to reduce the impact of deviating the model drastically from sudden exposure to unseen datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Three commonly used evaluation metrics, precision, recall, and F1- score (Equation 1-3), were applied to gauge the performance and bias of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Precision calculates the percentage of correctly identified location names (noted as True Positives, TP) among all the location names predicted by the model, which combines both TP and False Positives (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Recall measures the percentage of correctly identified ones amongst all ground truth, which is the combination of TP and False Negatives (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' F1-score is the harmonic mean of precision and recall, providing a comprehensive metric to evaluate model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = �� ����� (Equation 1) 𝑅𝑒𝑐𝑎𝑙𝑙 = �� ����� (Equation 2) 𝐹1–𝑠𝑐𝑜𝑟𝑒 = 2 ∗ ���������∗ ������ ���������� ������ (Equation 3) The outputs of BERT models are at word-piece level and they are concatenated using the special prefix ‘##’ and the word-level labels are assigned base on the starting word-piece of the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The evaluation metrics are based on ‘per-token’ scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Additionally, location name entity consists of two types of labels (B-LOC and I- LOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In order to gauge the comprehensive performance of the model on toponym recognition, the evaluation metrics were calculated using a micro average approach, which computes a global average of precision, recall, and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It calculates the TP, FP and FN by counting the total number of TP, FP and FN under each class, namely, “B-LOC” and “I-LOC”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 3 Results and Analysis The first step of the experiment targeted at determining the optimal pretrained parameters for BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' We hypothesize that larger models outperform smaller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' To verify this hypothesis, the performance of the models initialized with ‘bert-base-cased’ and ‘bert-large-cased’ with a linear classifier stacked on top were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The results are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 1: Evaluation results for testing on different pretrained parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' BERT Model Classifie r Precisio n Recal l F1- score bert-base-cased Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='902 bert-large- cased Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='917 These two models were trained with CoNLL2003-Train and evaluated with CoNLL2003-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Compared to ‘bert-base-cased’, the precision of the prediction increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='900 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='934 by using ‘bert-large-cased’ while the recall almost remained static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The F1-scores showed that ‘bert-large-cased’ rendered better results which is in conformity with the original BERT paper [12] and validated our initial hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Therefore, ‘bert-large-cased’ was harnessed in all the follow-up experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The second step of the experiments aimed to measure the influence of the training data and determine the optimal classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The model performances were evaluated using two different datasets, CoNLL2003-Test and Harvey2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' We hypothesize that (a) the model with CNN1D classifier yield better results and (b) models trained with larger datasets perform better in placename recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 2 and Table 3 list the evaluation metrics of all the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 2: Evaluation results with CoNLL2003-Test dataset for testing on training data variation and classifier types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Training Data Classifier Precision Recall F1-score CoNLL2003 Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='917 CoNLL2003 MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='907 CoNLL2003 CNN1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='921 Combined Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='866 Combined MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='912 Combined CNN1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='929 Table 3: Evaluation results with Harvey2017 dataset for testing on training data variation and classifier types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Training Data Classifier Precision Recall F1-score CoNLL2003 Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='847 CoNLL2003 MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='846 CoNLL2003 CNN1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865 Combined Linear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='703 Combined MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='685 Combined CNN1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='781 The “CoNLL2003” under the Training Data column means CoNLL2003-Train dataset and the “Combined” represents the dataset merging CoNLL2003-Test, Wiki3000 and WNUT2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In Table 2, when models were trained with CoNLL2003-Train, the one with a simple linear classifier produced the best precision (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='934), and the one with CNN1D produced the best recall (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='920) and F1-score (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='921).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' MLP performed the worst among the three classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' When models were trained with a combined dataset, the model with CNN1D outperformed the rest in all three metrics with precision equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='942, recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='916, and F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The one with a linear classifier produced the worst results with an F1- score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In Table 3, when models were trained with CoNLL2003-Train, the one with the CNN1D classifier outperformed the rest with precision equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='898, recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='835, and F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' When models were trained with a combined dataset, the model with CNN1D successfully defended its trophy by rendering precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='941, recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='668, and F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The models with MLP worked slightly worse than the ones with linear classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The above elucidation certifies the hypothesis that models with CNN1D generate the optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It also shows that more complicated classifiers like multi-layer perceptron do not necessarily render better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, when viewing Tables 2 and 3 contemporaneously, the results from training with different datasets, the metrics indicated that the model trained with the combined dataset generally performed worse than the ones trained with merely CoNLL2003- Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This phenomenon contradicts the hypothesis that models trained with larger datasets perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' After scrutinizing the dataset used for training, we noticed some inconsistencies in the labeling criteria of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Some examples are listed in Table 4 and the unexpected phenomenon can be interpreted by the heterogeneity of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 4: Examples of different labels across the datasets used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Example Entity Dataset CoNLL200 3 Wiki300 0 WNUT201 7 "Canadian" B-MISC O B-LOC "Planet" O O B-LOC "east" O O B-LOC "orchard" "academy" B-ORG/ I-ORG O B-LOC/ I-LOC "earth" O N/A B-LOC It can be seen from Table 4 that the word “Canadian,” which is labeled as “B-MISC” (beginning of a miscellaneous name), is identified as “B-LOC” (beginning of a location) in the WNUT2017 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The words “Planet”, “east,” and “earth” are misclassified as locations in the WNUT2017 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The phrase “orchard academy,” regarded as an organization under the CoNLL2003 criteria, is also labeled as a location entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In this case, combining several heterogeneous datasets can be considered adding some helpful unseen samples to the original training data while introducing a substantial amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [13] experimented on several deep learning models when trained with noisy data and claimed that the CNN model is more resilient to noise than MLP and linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The trend of performance change shown in Tables 2 and 3 when trained with different datasets is in accordance with this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It is noticeable that the models experience an increase in precision and a drastic decrease in recall when trained with a combined dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This incident can as well be triggered by noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Since deep learning models attempt to learn the underlying patterns of the training data, the existing noise will confuse the model, resulting in a fewer number of positive predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This might result in an increase in precision and a decrease in recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Based on the observation and interpretation above, the BERT model initialized with ‘bert-large-cased’, stacked with a CNN1D classifier and fine-tuned with CoNLL2003-Train was selected as the finalized TopoBERT module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 5 shows a comparison between TopoBERT and five other models and tools based on the Harvey2017 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Table 5: Evaluation results with Harvey2017 dataset for comparing TopoBERT with other existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Model Precisio n Recal l F1- score Stanford NER (broad location) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='548 SpaCy NER (broad location) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='366 BiLSTM-CRF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='649 DM_NLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='703 NeuroTPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='728 TopoBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865 The SpaCy version v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='0 is used with model “en_core_web_sm” loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Broad location indicates that we include entities in both LOCATION and ORGANIZATION for Stanford NER, and we include entities in the types of LOC, ORG, FACILITY, and GPE for spaCy NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Evaluation results show that TopoBERT prevailed in the competition with precision equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='898, recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='835 and F1-score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This result outperformed other baseline models by at least 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' TopoBERT has been developed as a ready-to-use module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The output data of TopoBERT includes word labels and confidence of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It complies with JSON file format for ease of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The source code has been uploaded to GitHub and can be accessed with the link: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='com/SPGBarrett/gearlab_topobert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 4 Discussion This paper presents a geoparsing framework and breeds a plug and play toponym recognition module which can facilitate spatial analysis based on social media or news media data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Figure 7 shows a practical application of this framework in locating Twitter posts under fine-grained topics during hazardous events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The study area is the State of Florida, and the dots in multiple colors displayed on the map are tweets posted during Hurricane Irma harvested by Twitter developer API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The locations of those tweets without geotags are retrieved by running TopoBERT and google geocoding service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The module also enjoys the potential of being used for location name detection for news media to pinpoint the discussed topics [14,15] and help to identify fake news [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Figure 7: Toponym recognition applied to locate Twitter posts during disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This paper concentrates mainly on designing a novel architecture of a reliable and versatile module for toponym recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, the performance enhancement can continue by addressing the following issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' First, the models are trained and evaluated based on well prepared datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This can be regarded as a best-case scenario compared to real life situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Place name usage can be highly ambiguous and random, especially within social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Typos are extremely common which might cause out-of-vocabulary words in language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Place name abbreviations such as “Boulevard” and “blvd”, “Drive” and “Dr.”, “Street” and “St.” and so forth are frequently utilized interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' People might unconsciously ignore the correct upper-case and lower-case usage, such as “college station” and “College Station”, “mexico” and “MEXICO”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Meticulous data preprocessing methods can be incorporated to tackle this problem in order to achieve better overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Second, several rule-base approaches can be leveraged to further boost the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Enlightened by the success of hybrid models [9], sets of grammar rules based on the composition of nouns, determiners, adjectives, conjunctions, numbers and possessive ending can be designed [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Additionally, commonly used gazetteers such as OpenStreetMap and GeoNames can be used as extra named entity matching criteria which will enhance the True Positives of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Regional criteria can be appended to the model while identifying place names by making country name, state names, county names, or bounding boxes as input variables of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This will allow the model to add constraints during processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The top-N words from word embedding models [9,35], which are not place names, can be applied to filter words during data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This will to some extent eliminate the False Positives of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Third, due to the data-hungry nature of deep learning, data availability and quality are topics being inevitably discussed when large complicated deep learning models are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' It is common knowledge in the deep learning world that larger datasets lead to better generalizability and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' However, this statement fails to hold true in this paper due to the fact that the larger datasets are derived from several distinguished smaller datasets labeled under their own unique regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Therefore, there is an urgent need to define criteria and build unified datasets for toponym recognition model training, evaluating and benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The dataset can be manually modified based on existing datasets and augmented using rule-based methods, gazetteers or Generative Adversarial Network [18,19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Fourth, fine-tuned language models can be few-shot or zero-shot learners, which means that the models can be applied directly to certain downstream tasks with very little or even no further training [21,22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This is because advanced language models can better capture the meaning of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' This claim is also underpinned by the result of this paper which leverages BERT to boost the module capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Therefore, incorporating gigantic models such as GPT- 3 [24] might lead to another round of performance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 5 Conclusion To further enhance the performance of toponym recognition by better understanding natural language, TopoBERT, which incorporate pretrained language model, BERT, is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Experiments on the pretrained parameters, training dataset combinations, and model architecture reveal the following findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' First, the toponym recognition model performance is sensitive to the architecture of pre-trained language models and classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The models initialized with a larger-structured BERT model (“bert- large-cased”) show an advantage over the models initialized with a basic BERT model (“bert-base-cased”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' More complicated classifiers like MLP do not necessarily win over simple linear classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Second, increasing training data size produces worse results, especially for the recall, due to data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The model trained with single dataset, CoNLL2003-Train, and stacked on top with a CNN1D classifier renders the optimum results both on CoNLL2003-Test and Harvey2017 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Finally, the developed TopoBERT module outperforms existing models in Hurricane Category IrmaRouteLine Florida_Census_ Tract_2019 0 Human_Help_Florida Animal HelpFlorida 3 Infrastructure Florida ShelterFlorida recognizing place names in texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The clinched TopoBERT with the optimal model architecture and training strategy produces reliable toponym prediction and achieves F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='865 on Harvey2017 dataset, which surpasses other prevailing models or tools by at least 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In nutshell, the discoveries of this paper contribute in determining the optimal model structure on toponym recognition tasks and urges a large standardized dataset labeled with unified regime to support model training and benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' A plug and play module is implemented and open sourced to support pertinent applications and similar research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' ACKNOWLEDGMENTS The research is supported by a project funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' National Science Foundation: Reducing the Human Impacts of Flash Floods Development of Microdata and Causal Model to Inform Mitigation and Preparedness (Award No.' metadata={'source': 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Transformers for Language Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [13] David Rolnick, Andreas Veit, Serge Belongie, and Nir Shavit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Deep Learning is Robust to Massive Label Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='10694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [14] Youjie Zhou and Jiebo Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Geo-location inference on news articles via multimodal pLSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=" In MM'12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The proceedings of the 20th ACM international conference on multimedia, co-located with ACM multimedia 2012, October 29- November 2, 2012, Nara, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Association for Computer Machinery, New York, 741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1145/2393347.' metadata={'source': 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Research and Development in Information Retrieval : August 9-13, 2015, Santiago, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' ACM, New York, 935–938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1145/2766462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='2767815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [16] Kai Shu, Xinyi Zhou, Suhang Wang, Reza Zafarani, and Huan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' The role of user profiles for fake news detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In ASONAM 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining : FAB 2019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' FOSINT-SI 2019,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In 2015 IEEE International Conference on Pervasive Computing and Communication Workshops (PerCom Workshops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' IEEE, 75–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1109/PERCOMW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='7133997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [18] Connor Shorten, Taghi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Khoshgoftaar, and Borko Furht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Text Data Augmentation for Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Journal of big data 8, 1, 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='1186/s40537-021-00492-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [19] Rui Cao and Roy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' HateGAN: Adversarial Generative-Based Data Augmentation for Hate Speech Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In Proceedings of the 28th International Conference on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' International Committee on Computational Linguistics, Stroudsburg, PA, USA, 6327–6338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='coling-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [20] Steven Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Feng, Varun Gangal, Jason Wei, Sarath Chandar, Soroush Vosoughi, Teruko Mitamura, and Eduard Hovy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' A Survey of Data Augmentation Approaches for NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [21] Jason Wei, Maarten Bosma, Vincent Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Zhao, Kelvin Guu, Adams W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Yu, Brian Lester, Nan Du, Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Dai, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Finetuned Language Models Are Zero-Shot Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [22] Mitchell Wortsman, Gabriel Ilharco, Jong W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Kim, Mike Li, Simon Kornblith, Rebecca Roelofs, Raphael G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Lopes, Hannaneh Hajishirzi, Ali Farhadi, Hongseok Namkoong, and Ludwig Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Robust fine-tuning of zero- shot models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [23] Pushpankar K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Pushp and Muktabh M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Train Once, Test Anywhere: Zero-Shot Learning for Text Classification.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='edu/~manning/papers/gibbscrf3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='pdf [33] Zhiheng Huang, Wei Xu, and Kai Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Bidirectional LSTM-CRF Models for Sequence Tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' [34] Chunping Ma, Huafei Zheng, Pengjun Xie, Chen Li, Linlin Li, and Luo Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DM_NLP at SemEval-2018 Task 8: neural sequence labeling with linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' In Proceedings of The 12th International Workshop on Semantic Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' Association for Computational Linguistics, Stroudsburg, PA, USA, 707–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FRT4oBgHgl3EQfujgS/content/2301.13631v1.pdf'} +page_content='org/10.' metadata={'source': 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Physics (THEP), +Johannes Gutenberg University, D-55099 Mainz, Germany +bDepartment of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A. +Abstract +We establish refactorisation conditions between the subleading O8-O8 contributions to +the inclusive ¯B → Xsγ decay suffering from endpoint divergences and prove a factorisa- +tion theorem for these contributions to all orders in the strong coupling constant. +This +allows for higher-order calculations of the resolved contributions and consistent summation +of large logarithms, consequently reducing the recently found large-scale dependence in these +contributions. We implement the concept of refactorisation in a heavy flavour application +of SCET, which includes nonperturbative functions as additional subtlety not present in +collider applications. +1 +arXiv:2301.01739v1 [hep-ph] 4 Jan 2023 + +Contents +1 +Introduction +3 +2 +General setup +4 +2.1 +Hard matching +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +3 +Bare factorisation theorem +7 +3.1 +B-type current (direct) contribution . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +3.2 +A-type (resolved) contribution +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Refactorisation of the endpoint contribution +12 +4.1 +Refactorisation at leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2 +Bare refactorised factorisation theorem . . . . . . . . . . . . . . . . . . . . . . . . +17 +4.3 +Refactorised factorisation theorem after renormalisation . . . . . . . . . . . . . . . +17 +5 +Summary and Outlook +20 +2 + +1 +Introduction +There has been a general belief that soft-collinear factorisation at subleading power in ΛQCD/mb +expansion is well established for inclusive B−decay modes such as ¯B → Xsγ, ¯B → Xsℓℓ, or +¯B → Xuℓ¯ν [1] – in contrast to exclusive B decays where factorisation theorems do not exist at +the subleading power in general. There are two types of subleading contributions to the inclusive +¯B → Xsγ decay, direct and resolved ones. In the latter, the photon does not directly couple to an +effective electroweak vertex, but they contain subprocesses in which the photons couple to light +partons instead. These subleading corrections are nonlocal in the endpoint region, and they stay +nonlocal even in the region where the local heavy mass expansion is applicable. In this sense, +they represent an irreducible uncertainty of this decay mode. Analogous subleading contributions +exist in the inclusive ¯B → Xsℓℓ decay but not in the inclusive ¯B → Xuℓ¯ν decay because, in this +case, the leptons can couple to light partons via the W vector boson only. +The first systematic analysis of resolved contributions to the inclusive ¯B → Xsγ decay [2,3] was +worked out in Refs. [4,5], the corresponding 1/mb contributions to the inclusive ¯B → Xsℓℓ decay +were discussed in Refs. [6,7], using soft collinear effective theory (SCET). Recently, the uncertainty +due to the resolved contribution was reduced with the help of a new hadronic input [8, 9]. But +these resolved contributions still represent the largest uncertainty in the inclusive ¯B → Xsγ decay. +Moreover, a large scale dependence and also a large charm mass dependence were identified in +the lowest order result of the resolved contribution, which calls for a systematic calculation of αs +corrections and renormalisation group (RG) summation [9]. A mandatory prerequisite for this +task is an all-order in the strong coupling constant αs factorisation formula for the subleading +power corrections. +The factorisation of resolved contributions introduces a new ingredient, namely an anti- +hardcollinear jet function [4], typically referred to as a radiative or amplitude-level jet function +in collider and flavour applications [10–18]. They are not represented by cut propagators as the +usual jet functions but as full propagator functions (both dressed by Wilson lines). But as al- +ready noticed in Ref. [4], the specific resolved O8 − O8 contribution does not factorise because +the convolution integral is UV divergent. The authors of Ref. [4] emphasised that there is an +essential difference between divergent convolution integrals in power-suppressed contributions of +exclusive B decays and the divergent convolution integrals in the present case, while the former +were of IR origin, the latter divergence of UV nature. However, a solution at the lowest order +was established by considering the sum of direct and resolved O8 − O8 contributions, which was +shown to be scale and scheme dependent by using a hard cut-off in the resolved contribution. But +the failure of factorisation does not allow for a consistent resummation of large logarithms. +In this paper, we identify the divergences in the resolved and in the direct contributions +as endpoint divergences by showing that also the divergence in the direct contribution can be +traced back to a divergent convolution integral. +Recently, new techniques were presented in +specific collider applications of SCET [19–27]. The so-called refactorisation conditions or endpoint +factorisation [22,23,25–27] allow for an operator-level reshuffling of terms within the factorisation +formula so that all endpoint divergences cancel out. In this work, we now implement this idea +in a flavour application of SCETI, which includes nonperturbative soft functions not present in +collider applications – often referred to as subleading shape functions [28,29]. +As a first step, we derive the matching of the hard function for the two operators involved +in the O8 − O8 subleading contributions. In the second step, we establish the bare factorisation +theorem for the direct and the resolved contribution at an operational level. Then we derive +3 + +the refactorisation conditions to all orders, leading us finally to the renormalised factorisation +theorem. We present all steps for the inclusive ¯B → Xsγ, but all the details can also be taken +over for the corresponding ¯B → Xsℓℓ case. +2 +General setup +The starting point for all calculations concerning the ¯B → Xsγ decay is the weak effective +Lagrangian defined at a scale µb parametrically equal to the b-quark mass µb ∼ mb. The weak +effective Lagrangian is obtained from the SM Lagrangian after integrating out the heavy particles +like the heavy gauge bosons and the top quark. We use the convention of Ref. [30]. Assuming +Standard Model CKM unitarity, with λq = VqbV ∗ +qs and λu + λc + λt = 0, the effective Hamiltonian +may be written as +Heff = GF +√ +2 +� +q=u,c +λq +� +C1 Oq +1 + C2 Oq +2 + C7γ O7γ + C8g O8g + +� +i=3,...,6 +Ci Oi +� +. +(1) +Here we concentrate on the O8 operator: +O8g = − gs +8π2 mb ¯sσµν(1 + γ5)Gµνb . +(2) +Our sign convention is that iDµ = i∂µ+gs taAa +µ+e qfAµ, where ta are the SU(3) colour generators, +and Qf is the electric charge of the fermion in units of e. We consider the CP-averaged ¯B → Xsγ +photon energy spectrum in the endpoint region where Mb − 2Eγ = O(ΛQCD). Soft-collinear- +effective theory (SCET) offers the appropriate framework for this multi-scale problem. +The +kinematics of the decay is given as follows: the initial meson carries momentum pB, and it decays +into a photon with momentum q and a jet whose total momentum is PX. From pB − q = PX in +the B meson rest frame, we have 2MBEγ = M 2 +B − M 2 +X. Thus, the jet invariant mass MX is much +smaller than the photon energy Eγ and jet energy EX. We set PX⊥ = 0 and choose reference +vectors n2 = n2 = 0, v2 = 1, such that n + ¯n = 2v and nn = 2. Choosing +qµ = Eγ¯nµ +and +pµ +B = MBvµ , +(3) +we find MB = ¯nPX and MB = nPX + 2Eγ or equivalently +P µ +X = (MB − 2Eγ)nµ + MB¯nµ . +(4) +Thus, there is only one independent kinematical variable in the ¯B → Xsγ decay. One may choose +the photon energy Eγ or nPX = MB − 2Eγ. +Three dynamical scales describe the endpoint region, a hard scale of O(MB), an intermedi- +ate (anti-)hardcollinear scale of O( +� +MBΛQCD), and a soft scale of O(ΛQCD). The expansion +parameter in our present analysis is defined as λ2 = ΛQCD/MB1. The photon can be treated as +anti-hardcollinear. The hadronic final state factorises into a hardcollinear jet and soft wide-angle +radiation. Since the soft modes have parametrically smaller virtuality than the hardcollinear +1Alternative convention often used in the literature is to define λ = ΛQCD/MB. +4 + +modes, the problem at hand is described by the SCETI setup [31–34]. Using a shorthand nota- +tion a ∼ (na, ¯na, a⊥) to indicate the scaling of the momentum components in powers of λ, we +have: hard momentum scales like phard ∼ (1, 1, 1)mb, a hardcollinear one phc ∼ (λ2, 1, λ)mb, an +anti-hardcollinear region phc ∼ (1, λ2, λ)mb and a soft momentum psoft ∼ (λ2, λ2, λ2)mb. +The first step in the derivation of a factorisation theorem is hard matching. We have to match +the electroweak operator onto SCET. We will see that the direct contribution is represented by a +next-to-leading power (NLP) B-type current in SCET, i.e. power-suppressed current composed of +two collinear building blocks. The resolved contribution is represented by a time-ordered product +of a leading-power (LP) A-type current with a subleading L(1) +ξq Lagrangian (see Ref. [19, 35–37] +for a precise definition of the different types of currents). +In the second step, we integrate out the hardcollinear fields, which lead to the appearance of +jet functions. The latter are, technically speaking, matching coefficients of SCET on the pure +soft effective field theory. Kinematics forbids the emission of anti-hardcollinear partons. Thus +anti-hardcollinear fields have to be integrated out at the amplitude level. The physics in the +anti-hardcollinear direction is similar to the threshold Drell-Yan expansion, where kinematical +constraints forbid hardcollinear emissions into the final state. For more details on SCET NLP +factorisation and resummation in the threshold Drell-Yan, see Refs. [13,14,20,38]. In the hard- +collinear sector, the formalism resembles, for example, the thrust factorisation, where NLP jet +functions are defined at the cross-section level [15,39–42]. +2.1 +Hard matching +The electroweak operator O8 matches onto two possible SCET operators. First, we consider the +A-type current, which enters the resolved contribution. Within the present section, the SCET +operators are always given before the soft decoupling transformation [32] is performed. +The +A-type SCET operator is given by +OA0 +8g (0) = χhc (0) /n +2γµ⊥Aµ +hc⊥ (0) (1 + γ5) h (0) , +(5) +where h is the heavy quark field, and the SCET building blocks are hardcollinear gauge-invariant +due to the introduction of hardcollinear Wilson lines W. The fermionic building block is χhc = +W † +hcξ and gluon field is +Aµ +hc⊥ = W † +hc [Dµ +hc⊥Whc] = Aaµ +hc⊥ta. +(6) +Note that the colour and Dirac structure for the A-type operator is uniquely fixed. +For the +matching, we can use the partonic QCD amplitude b (pb) → s (ps) g (r), where the momenta pb is +hard, ps anti-hardcollinear and r hardcollinear. The matching condition is given by +GF +√ +2λt C8g ⟨Q8g⟩ = CA0 (mb) +� +OA0 +8g +� +. +(7) +The brackets ⟨ ⟩ indicate that the matrix element of the operators is considered. At leading order +in αs we find +CA0 +LO (mb) = m2 +b +4π2 +GF +√ +2λqC8g . +5 + +The B-type current which enters the direct contribution is the following SCET operator:2 +OB1 +8g (u) = +� +dt +2πe−iumbtχhc (t¯n) γν⊥ Qs Bν +hc⊥ (0) γµ⊥ Aµ +hc⊥ (0) (1 + γ5) h (0) , +(8) +with Qs as the electric charge of the strange quark in units of e and the electromagnetic gauge- +invariant transverse photon field +Bν +hc⊥ = e +� +Aν +⊥ − ∂ν +⊥ +n∂ nA +� +. +(9) +We note that beyond the LO, a second operator with an alternative Dirac structure γµγν appears. +These operators do not mix under renormalisation [43]. We checked by explicit computation at +the one-loop order that the matching coefficient for the operator with γµγν structure does not +develop any endpoint divergence. +We use the partonic QCD amplitude b (pb) → g (r) s (ps) γ (q) to fix the matching coefficient. +Here the momenta are pb hard, r and ps hardcollinear and q anti-hardcollinear. On the QCD +side, a time-ordered product of the operator O8 and of the QED current, +LQED,q (x) = eq Aµ (x) q (x) γµq (x) , +(10) +is needed. The matching condition at the leading order in QED is given by +GF +√ +2λtC8g i +� +d4x ⟨T [O8g (0) , LQED,b (x) + LQED,s (x)] ⟩ = +� 1 +0 +duCB1 (mb, u) +� +OB1 +8g (u) +� +. +(11) +We do not consider QED corrections. At leading order in αs, we find that only the QED current +with an s quark contributes, and we arrive at +CB1 +LO (mb, u) = (−1)u +u +m2 +b +4π2 +GF +√ +2λt C8g = (−1)u +uCA0 +LO (mb) , +(12) +where we use hardcollinear momentum conservation ¯ns + ¯nr = mb and introduce hardcollinear +momentum fraction u = ¯nps +mb and u = 1 − u. +Finally, we compare the different kinematics of the A- and B-type currents in Figure 1. The +external s-quark carries hardcollinear momentum. Therefore the intermediate propagator is hard. +This situation is represented in SCET by the B-type current. When the momentum of the external +s-quark tends to zero, the propagator becomes anti-hardcollinear and cannot be integrated out – +it must be reproduced by a dynamical field in the low energy EFT. This situation is represented +in SCET by the time-order product of subleading Lagrangian and the A-type current. +The +degeneracy in the EFT description is the reason why the SCET develops divergencies in the +convolution integrals. +2Note that this operator is equal to J2 (τ) in Ref. [43] (eq. 16), with J2 (τ) = 2mbOB1 +8g (τ). +6 + +AOBHicrVdbc9tEFHZLgGItPDIi +4YkQ0u3RpLtOHnwTJu205bJNGlLmzJR8KzktaXJ6pLVyq3Z2Vce+S +W8AY/0iT/Bv+GsLra8crnMoLGd1Tnf+c5lzx4pbkKDlJvmn5cuv7Px +7nvX/mg/eFHmx9/cvXapy/SOGMe7FNGYvXZwSGkTkOQ84JS8TR +nDoUnLint9V+pMZYWkQR9/yeULOQjyNgkngYQ6i0bWNLx2XTINITM +LJKBEigmZR1OGE1+2DWOpzEVfyevWwEQD8wboQAtySiZcBYKuiw +5ULM40RwC/Eu4iDdWaCxS6jYdjw/GN3dlmDYVF6MUtDwbqGpKYTjY +WrckSPhJIQlhicBF1vSaOeomRgTL2ad1McJGXoB8yhBhQgyo2Q8hNB +LSPADGXa5H3jnUswghn9JQMKEz+scezkHygMcbh+N9rYR7YyHJrCq ++AtW0Dgh5j5ELw7l9+K6dUPl8DowLmSJR/Zs2V4dv/4Cj8P5x0l1 +vLgqnPoe4o7qJYOb/IMDs38C3oRtqu7ZdbrASCZ/g2RxEqlWHFq +dvtKhRZWg7VgImqUjouZ8L089DQYk2HudWmvtg3lqSd+zOocFKsF +ygbfNhoVoZQ9d4UhyG+sdwLtSxUW5UJ6n23ZnSrCQr4dNRHkbeaI +WTHLKayQwqoTo5YXGCp5jHrL2SVAXu7Fb5+V+v+N/OVSkRQRGYte +DSBdB2HI9s50XJK56oapE5c17Sx3NjmogVGyvQ6Jx7ai3q/t8Loyub +pkdM7+M5sIqF1ut8joeXbvyhzOvSwkEfcoTtNTy0z4mcCMB9B5QJ ++lJMHeOZ6SU1hGOCTpmchHmjR2QDI2JjGDb8SNXFq3EDhM03noAlJ1 +fKrlHCd7jTjk70zEURJBjXwCkeTjBo8NtR8NMYBIx6nc1hgjwUQq ++H5mGPwxRd8ZJCUj4ZyzZcTkReTH0FVTMwTSV6jfx8SiVmu7tKpI +oHfwGNI40XczGUhTz70iulE6EGeUBi19JLQrA3gG+ah7ckbKhf1zT +P16jP6rpldtVwFECIT3RzY6SR5qVrq7rv2mw3j+JYhauBAZn9R6hH +BsHxtCwGhZhdv9E4bPRidQ1bqFwdRtMn719i+j9hc7h5DUXJNTOIQ +n7JO792SxGldAJVp1ZTjx4al1JmrjZgtSUE3jMkE7vPWFmMYoVNG +SCQFm0LUcLKQCeMC/som0qUZPKmfPjiQwuoi9bH6GgzI5guHiq/Un +Pr3QO2Bzk0Z9u3UP7RyLwaxO71kd3rIru7p6MOlmGZEBJ8m0yHav7 +UgLbVA6oBsvu2Dn0Kh6tMch8qgSxTr0XBtsTZfSCzbUW4JgNV3Cq43 +V20t56tDvwbviOGo+kiCdtCgx7q9dYyrkL7UD5LfRvujzOW0Aq30 +XdPrJsfV8LyhWo3bMgQnNdCb8jlMJoqFzbakdQ0/OjZ8+eapsHsEj +HYWrtYK1O4BK7gNSd3z8sIKYUL0B6tr6CXQpzC+pjkZ+dAqzQpifk +FV03s0auOjwJjY/HA1ilVwTGxdbo6FL6Ro8vD3D4wzrBpV4jYU3x5 +EOz2XrsyQNcDkVmujlyNAsarNkvdW6Ci3HStOGwQHT4EpUIFehsPEa +UrXCukKmaMsVeXvNBQhMlQkSjm1JW96G6v7W4n7lCzHT2KFSP0 +vIBLSIJeA2QaICbymHtqar+CwoTEMJ7j6W/5TQXL+yOtdvpPult3T4 +o34CutD5vfdG63rJag9bt1sPWcet5y9v4aePXjd83mz+uPnz5i+b +vxXQy5dKm89aK9fm78ADbYEJA= +h/hc +b +�hc +O8 +ghc +shc +Figure 1: The full theory LO diagram which induces an endpoint divergence in SCET, see text. +3 +Bare factorisation theorem +The derivation of the factorisation theorem follows the standard approach. [44, 45]. +We first +perform the soft decoupling transformation [32], but we do not use a new notation for the hard- +collinear fields after decoupling. +The decay rate is obtained from the imaginary part of the +current-current correlator. The states factorise and thus allow taking matrix elements separately +in hardcollinear, anti-hardcollinear and soft sectors. +The common to both A- and B-type contributions is the anticollinear matrix element of the +photon. It is given by a discontinuity of the photon propagator +−gµν +⊥ e2 Jγ +� +q2� += +1 +2πi Disc[ i +� +d4xeiqx ⟨0| T [Bµ +hc⊥ (x) , Bν +hc⊥ (0)] |0⟩ ] +(13) += −gµν +⊥ e2 δ +� +q2� +θ +� +q0� += −gµν +⊥ e2 δ+ � +p2� +(14) +Since we are only interested in the photon-final state, the above expression is exact to all orders +in perturbation theory. +3.1 +B-type current (direct) contribution +There are several functions entering the factorisation formula of the direct contribution with the +B-type current. The soft function – the leading power shape function – is defined as +S (ω) = +1 +2mB +� +dt +2πe−iωt ⟨B| h (tn) Sn (tn) S† +n (0) h (0) |B⟩ , +(15) +or with open indices [46,47]3 +1 +2mB +� +dt +2πe−iωt ⟨B| [hα (tn) Sn (tn)]i +� +S† +n (0) hβ (0) +� +j |B⟩ = δij +2 Nc +�1 + /v +2 +� +βα +S (ω) . +(16) +3We use Greek for spinor indices and Latin for colour indices. +7 + +The hardcollinear jet function is a genuine NLP object. In analogy to the LP jet function, we +define it as a vacuum matrix element of a product of hardcollinear fields +J +� +p2, u, u′� += (−1) +2Nc +1 +2π +� +dtdt′ +(2π)2 d4x e−imb(ut−u′t′)+ipx +(17) +Disc +� +⟨0| tr +�1 + /v +2 +(1 − γ5) /Ahc⊥ (x) γν +⊥χhc (t′¯n + x) χhc (t¯n) γν⊥ /Ahc⊥ (0) (1 + γ5) +� +|0]⟩ +� +. +The field operators are time- or anti-time-ordered according to the Keldysh formalism.4 +The +trace is taken both with respect to colour and spinor spaces. Using projection properties of the +hardcollinear fields and 2/v = /n+ + /n− the Dirac algebra in eq.(17) can be simplified to +J +� +p2, u, u′� += (−1) +2Nc +1 +2π +� +dtdt′ +(2π)2 d4x e−imb(ut−u′t′)+ipx (d − 2)2 +(18) +Disc +� +⟨0| tr +�/¯n +4(1 − γ5)Aµ +hc⊥ (x) χhc (t′¯n + x) χhc (t¯n) Ahc⊥ +µ +(0) (1 + γ5) +� +|0⟩ +� +where, as mentioned before, the trace is also applied in the colour space. +With these definitions, we find the bare factorisation theorem for the direct contribution +dΓ +dEγ += NB +� 1 +0 +duCB1 (mb, u) +� 1 +0 +du′CB1∗ (mb, u′) +� Λ +−p+ +dωJ (MB (p+ + ω) , u, u′) S (ω) +(19) +with the prefactor +NB = [(2π)] [e2Q2 +s] [ +1 +(2π)3 2 Eγ +E2 +γ 4π] = e2Q2 +s +Eγ +2π +(20) +The three pieces of the prefactor correspond to the phase space factors of the photon, to its +charges and to the redefinition of the jet function with a 2π factor. +Finally, we prove to all orders in αs that the jet-function is symmetric in u and u′ up to +complex conjugation: +J(p2, u, u′) = J∗(p2, u′, u). +(21) +This can be read off from the factorisation theorem of the direct contribution. The photon energy +spectrum is real. The leading power shape function is also real to all orders. This can be shown +by complex conjugation of Eq.(15) and by using translation invariance [4]. Then the jet function +inherits the symmetry property given in Eq.(21), from the product of the Wilson coefficients, +CB1 (mb, u) CB1∗ (mb, u′), in the convolution integral. An anti-symmetric part of the jet function +would cancel out in the convolution integral. We emphasise that this property is also valid when +the other B-type operator with the reversed Dirac structure is present. In particular, the sum of +the two mixed terms has this property. In the latter terms, the reduction of the Dirac structure +leads to (4 − d) (d − 2), and hence these terms vanish for d = 4. +The symmetry property is crucial for the refactorisation because it implies that no double +subtraction regarding the variables u and u′ is needed in the B-type (direct) current contribution. +This can be seen in the following way. We showed above that the integrand of the convolution +integral of the Wilson coefficients and the jet function in the two variables u and ¯u is real, so the +4For a brief summary, see appendix of Ref. [48]. +8 + +complete integrand is symmetric in u and u′. Then the subsequent rearrangement is possible (we +here only write the convolution variables u and u′): +� 1 +0 +duCB1 (u) +� 1 +0 +du′CB1∗ (u′) J (u, u′) = 2 +� 1 +0 +duCB1 (u) +� 1 +u +du′CB1∗ (u′) J (u, u′) . +(22) +As the endpoint divergence manifests for small u and u′, we need to ensure that only the last +integral over u is rendered finite by an appropriate subtraction. +At the leading order, the jet function is real, and we find that the jet function is symmetric +in u and u′. Explicitly, we find using the dimensional MS regulator (µ2ϵ → µ2ϵ exp(γEϵ)/(4π)ϵ): +J +� +p2, u, u′� += CF +αs +4π mb +θ(p2) A(ϵ) δ(u − u′)u1−ϵ(1 − u)−ϵ +�p2 +µ2 +�−ϵ +, +(23) +with +A(ϵ) = (2 − 2ϵ)2 (1 − 1/2 ϵ) Γ(1 − ϵ)−1 exp(γEϵ) = 4 − 10ϵ + O(ϵ2) . +(24) +We compute the convolution integrals explicitly5 using this leading order result for the jet +function and also the hard function at leading order, Eq.(12), +dΓ +dEγ +|B = 2NB +��CA0 +LO (mb) +��2 � 1 +0 +du ¯u +u +� 1 +u +du′ ¯u′ +u′ +(25) +CFA(ϵ) +αs +(4π) mb +� Λ +−p+ +dω S (ω) +�mb(p+ + ω) +µ2 +�−ϵ +u1−ϵ(1 − u)−ϵδ(u − u′) += NB +��CA0 +LO (mb) +��2 CF +αs +(4π) mb +� Λ +−p+ +dω S(ω) A(ϵ)B(3 − ϵ, −ϵ) +�mb(ω + p+) +µ2 +�−ϵ +, +where B(x, y) denotes the Beta function. We see that the divergence in the direct contribution +is now identified as an endpoint point divergence in the convolution integral of the hard and the +jet function in the u integration for u ≪ 1. +We emphasise that this endpoint divergence can be regularised within the dimensional regu- +larisation scheme6. This leads to additional poles after performing the convolution. Consequently, +due to endpoint divergences, the bare factorisation formula is already invalid for the d → 4 limit +at the leading order. +5Symmetry of the original integral implies that +� 1 +u du′δ(u − u′) = θ(0) with θ(0) = 1/2, for u ∈ [0, 1]. +6We note that we do not confirm the leading order result of the direct contribution of Ref. [4] in the dimensional +regularisation scheme. In the notation of Ref. [4] we get +F (a) +88 (Eγ, µ) = CF αs(µ) +4π +� mb +2Eγ +�2 � ¯Λ +−p+ +dω +�2 +9 ln mb(ω + p+) +µ2 ++ 2 +9 +� +S(ω, µ) . +9 + +3.2 +A-type (resolved) contribution +For the resolved contribution with the A-type current, we start with the time-ordered product +OTξq = i +� +ddxT +� +Lξq (x) , OA0 +8g (0) +� += i +� +ddxT +� +qs (x+) Sn(x+) +� +Qs /Bhc⊥ + /Ahc⊥ +� +(x) χhc (x) , +χhc (0) S† +n(0)Sn(0)/n +2 /Ahc⊥(0) (1 + γ5) S† +n(0)h (0) +� +. +(26) +The operator in the hardcollinear sector contains only gluon fields. Hence the standard leading +power gluon jet function appears +−g2 +sδabgµν +⊥ Jg +� +p2� += +1 +2πi Disc +� +i +� +d4xeipx ⟨0| T +� +Aaµ +hc⊥ (x) , Abν +hc⊥ (0) +� +|0⟩ +� +. +(27) +At leading order we find the standard result Jg (p2) = δ+ (p2). +Besides photons, there are no energetic particles emitted in the anti-hardcollinear directions. +Thus, the anti-hardcollinear jet function is defined at the amplitude level: +OTξq = +� +dω +� +dt +2πe−itω [qs]α (tn) +� +J (ω) +�a νµ +αβ +Qs Bν +hc⊥ (0) Aµ +hc⊥ (0) [h (0)]β . +(28) +The anti-hardcollinear jet function can be decomposed as +� +J (ω) +�a νµ +αβ = J (ω) ta +� +γν +⊥γµ +⊥ +/¯n/n +4 +� +αβ +, +(29) +to all orders. The other structure γµ +⊥γν +⊥ does not appear as one can read off from the structure of +the T product in eq. (26) and the fact that the gluon and heavy quark fields are only spectators. +The Dirac structure can then be simplified at the level of the cross-section with the help of the +following relation: +� +γν +⊥γµ +⊥ +/¯n/n +4 +� +αβ +� +γµ +⊥γν +⊥ +/n/¯n +4 +� +α′β′ += (d − 2)2 +�/¯n/n +4 +� +αβ +�/n/¯n +4 +� +α′β′ +. +(30) +At leading order, the anti-hardcollinear jet function is given by +� +J (ω) +�a νµ +αβ = +ta +(ω + i ϵ) +� +γν +⊥γµ +⊥ +/¯n/n +4 +� +αβ +. +(31) +Having defined hardcollinear and anti-hardcollinear functions, we now focus on the soft sector. +The operatorial definition of the soft function in position space with open Dirac indices is +Sαβ,α′β′ (u, t, t′) = += g2 +s ⟨B| +� +h (un) (1 − γ5) +� +α′ +� +Sn (un) taS† +n (un) +� +S¯n (un) +� +S† +¯n (t′¯n + un) qs (t′¯n + un) +� +β′ +× [qs (t¯n) S¯n (t¯n)]α S† +¯n (0) +� +Sn (0) taS† +n (0) +� +[(1 + γ5)h (0)]β |B⟩ / (2mB) . +(32) +10 + +We can now plug in all the objects into the matrix element squared, and we find the resolved +contribution +dΓ +dEγ += NA +��CA0 (mb) +��2 � Λ +−p+ +dωJg (mb (p+ + ω)) +� +dω1 +� +dω2J (ω1) J +∗ (ω2) S (ω, ω1, ω2) , +(33) +with the prefactor +NA = NB ≡ N , +(34) +and the scalar soft function obtained after contracting spinor indices according to +S (u, t, t′) = (d − 2)2g2 +s ⟨B| h (un) (1 − γ5) +� +Sn (un) taS† +n (un) +� +S¯n (un) S† +¯n (t′¯n + un) +(35) +/n/¯n +4 qs (t′¯n + un) qs (t¯n) /¯n/n +4 S¯n (t¯n) S† +¯n (0) +� +Sn (0) taS† +n (0) +� +(1 + γ5) h (0) |B⟩ / (2mB) . +The soft function in momentum space, which appears in eq. (33), is obtained through the Fourier +transform of the position space expression according to +S (ω, ω1, ω2) = +� du +2πe−iuω +� +dt +2πe−itω1 +� dt′ +2πeit′ω2S (u, t, t′) . +(36) +As the NLP jet function in u and u′ variables, the soft function S (ω, ω1, ω2) is symmetric in +ω1 and ω2 up to complex conjugation: +S (ω, ω1, ω2) = S∗ (ω, ω2, ω2) . +(37) +This property stems from the fact that the gluon jet function is real to all orders. Thus, the +soft function inherits the symmetry property from the product of the anti-hardcollinear jet func- +tions, J (ω1) J +∗ (ω2) in the factorisation formula of the resolved contribution. Any anti-symmetric +part would cancel in the convolution integral. This symmetry property implies that within the +refactorisation, a double-subtraction regarding the variables ω1 and ω2 in the A-type current +contribution is not needed either. The symmetry implies that the integrand in the convolution +integral between anti-hardcollinear jet functions and soft function in the two variables ω1 and ω2 +is real and symmetric and allows for the following rearrangement of the convolution integral +� ∞ +−∞ +dω1 +� ∞ +−∞ +dω2J (ω1) J +∗ (ω2) S (ω1, ω2) = 2 +� ∞ +−∞ +dω1 +� ω1 +−∞ +dω2J (ω1) J +∗ (ω2) S (ω1, ω2) , +(38) +which is motivated by the fact in the resolved contribution. As we will see explicitly in section 4.1, +the convolution integral of the jet and shape function is logarithmically divergent for ω1,2 → ∞. +At leading order, we find the factorisation formula in case of the A-type current (resolved) +contribution:7 +dΓ +dEγ += 2N +��CA0 +LO (mb) +��2 � Λ +−p+ +dωδ (mb (p+ + ω)) +� ∞ +−∞ +dω1 +� ω1 +−∞ +dω2 +1 +(ω1 − iϵ) +1 +(ω2 + iϵ) S (ω, ω1, ω2) . +(39) +We keep the soft function unevaluated at this point since this is a nonperturbative object. For +ω1,2 ≫ ω, the soft function can be shown to be asymptotically constant, which leads to endpoint +divergence in the convolution integrals for large ω1,2 (see section 4.1). +7This confirms the leading order result of the resolved contribution of Ref. [4] when the asymptotic limit of the +soft function is not yet considered. +11 + +Figure 2: Scales relevant to refactorisation of the endpoint divergent contribution. The left part of +the diagram represents the standard hierarchy of three scales for SCETI. Near the endpoint, when +the momentum fraction u is no longer u ∼ O(1), i.e. u ≪ 1, we introduce additional, unphysical +scales which make it possible to factorise further objects appearing in the bare factorisation +theorem. +4 +Refactorisation of the endpoint contribution +We here state the three refactorisation relations, which are based on the fact that in the limits +u ∼ u′ ≪ 1 and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8−O8 contribution have the same +structure. The refactorisation relations are operatorial relations that guarantee the cancellation +of endpoint divergences between the two terms to all orders in αs. +The refactorisation conditions result from the overlap between soft and hardcollinear modes. +The hierarchy of scales near the endpoint is shown in Fig. 2. +We will refer to these overlap +modes as softcollinear modes. +They play a similar role as the z-SCET modes introduced in +Ref. [24] to prove the refactorisation of the B1-type matching coefficients. +The parameter z +corresponds to the momentum fraction u in the present analysis. On the one hand, we can think +of the softcollinear mode as a limit of hardcollinear mode when the large momentum fraction +tends to zero8. +On the other hand, the softcollinear modes can be understood as a limit of +the soft modes when the n+k momentum component becomes much larger than the remaining +components, mb ≫ n+k ≫ λ2mb. We want to emphasise that softcollinear and u-hardcollinear +modes are not physical but help introduce refactorisation. The softcollinear fields obey the same +projection properties and have the same transformation properties regarding gauge invariance as +their hardcollinear counterparts. +• Following Refs. [22–24], we find that in the limit u → 0, the matching coefficient can be +8Thus, they do not appear in the leading power problems, where only operators with a single hardcollinear +field in each direction occur. +12 + +2 +hard +B1 +CAO 7 +umb +u-hardcollinear +hardcollinear +J&S→J,3 +softcollinear +S↑ +softfurther factorised +� +CB1 (mb, u) +� += (−1)CA0 (mb) mbJ (umb) , +(40) +where �g(u)� only denotes the leading term of a function g(u) in the limit u → 0 and +without any higher power corrections in u ≪ 1. The function J (umb), which appears here, +is exactly the same radiative jet function (29) we introduced before in the context of A-type +contribution. +This refactorisation condition stems from the fact that in the limit u → 0, the amplitude +used in the matching of the B-type current can be represented by a time-ordered product +[24], +CB1 (mb, u) +� +OB1 +8g (u) +� ��� +u→0 = CA0 (mb) i +� +ddxe−i (nx/2) umb +� +T +� +L(1) +ξqsc (x) , OA0−u +8g +(0) +�� +, +(41) +of the leading power current OA0−u +8g +(0) = χu−hc(0) S† +n(0) Sn(0) /n +2 /Au−hc⊥(0) (1 + γ5) S† +n(0) h(0) , +equal to (5) up to a replacement of the hardcollinear fields by the u-hardcollinear fields, and +subleading Lagrangian +L(1) +ξqsc (x) = qsc(x+)S† +n(0) Sn(0) +� +Qs /Bu−hc⊥ + /Au−hc⊥ +� +χu−hc(x) + h.c. +(42) +The jet-function J (umb) appears after integrating out the u-anti-hardcollinear quark fields. +We note a close resemblance to the structure of the resolved contribution, where a similar +time-ordered product appears (see Eq. (26)). +• We find the new soft function �S (ω, ω1, ω2) which corresponds to the function S (ω, ω1, ω2) in +the limit ω1 ∼ ω2 ≫ ω. In this limit, we can consider the light soft quarks to be softcollinear +qs → qsc. In this function �S (ω, ω1, ω2) higher power corrections in ω/ω1,2 are neglected. +• In the limit, where the momentum fractions u → 0 and u′ → 0, the jet function +J (mb (p+ + ω) , u, u′) fulfills the following relation +� Λ +−p+ +dω �J (mb (p+ + ω) , u, u′) S(ω)� = +� Λ +−p+ +dωJg(mb(p+ + ω)) �S(ω, mbu, mbu′) , +(43) +where the brackets indicate that the u → 0 and u′ → 0 limits have to be taken and that the +hardcollinear quark fields in J are regarded as softcollinear fields, χhc → qsc in accordance +with (41). +It is crucial that the soft function �S (ω, ω1, ω2) appears both in the A-type contribution in +the limit ω1 ∼ ω2 ≫ ω and in the B-current term if one expands for small u and u′. +Before we proceed, let us comment on the structure of �S. In the asymptotic regime, where +ω1,2 ≫ ω, we can match the �S on the leading power shape function +�S (ω, ω1, ω2) = +� +dω′K(ω, ω′, ω1, ω2)S(ω′) . +(44) +13 + +AONHicrV +fNbtGEFZSt03Vuk3aYy9EbQNJsxFISrLsg4DYSZC +kMGInaeIUlisqZVIePnj5UqJutjX6rGnvkSB3toe +m2foLH8kcsm0KBDCFJYz38w3Mzs7pJ2Y+gk3zd+uX +P1g48OPr72SfvTzY/+L6jS9fJtGcueSFG9GIvX +JwQqgfkhfc5S8ihnBgUPJqXNxT+lPF4QlfhR+z5c +xOQ/wLPSnvos5iMY3Nk5HDpn5oZgG06lPiRTsgxn +DMebBvGWpmKvpU3rYGJBuYt0IEW5JRMufAt5HeRb +8uVmEex4BbiXcRBurNCY4dQsT1yPX98b1uCYV15OU +5Aw7uZpqQIxdT40COxSgmLDZcCbjIkY7RS3EhLgR +6yQejsnQ9ZlLCcpEkBklkyEJYr7MRYn/ExlaNvd89 +wKlBMPtUYC5BxziSHG8Y1L+aO4ad2S24h2JkNTig +XE+57IDsy1V5Vr5rUaRcZexJLj0f9ntlJmIL/gyh +4HyTdRswf+Zx2CMUdVGkyC/nmF0Y+A50Lm2X9tbJ +m0GJhEfwYok4CVXbDq1OX+nQqkrQoiwAzWrbHMyE5 +6ahJ/6EDFPWtb3aNpSmHnsRj8KMJFuvUDZw2GiRh1 +D06QwHAYb6R3CG1BFTNKrVKvzOg8d5bDZ+M0jLQ +pM5IUs1kAZVQXR+zKMYzCPWriRVgDu9VX5l9u3U +g4oz4zdiuxCAsGywEiKU8DsWzdzstSlT0Q1GNn +NFz31FLs6OaCGVbPCLhpDQa2sVzOkfG17fMjpleRn +1h5YutVn6djG9c+3U0idx5QELuUpwkZ5YZ83OBGfe +h+8D9PCExdi/wjJzBMsQBSc5FOgKlsQOSiTGNGNwh +N1Jp2ULgIEmWgQNI1fWJrlPCJt3ZnE/3zoUfxnOog +ZsRTefU4JGh5qkx8RlxOV3CArvMh1gN18Muxymbo +UlgaQ8MpFtuEYhe1G0FtQsRGmiVS/sYfHidR071a +RWOng16dRqOkiNpEim5fHslI6Ecwp91n0WmpRAPYA +/BUz4UDKmv5JSf+kQX9c0ivaKuA4hpCe6mbH8WPNS +leX9d/VvD4DSMWVAKDI3CfUI6NQ2NoWDWLYP7gVO +Hn41Opa5xM4eg2mD5/9xbRByvdiJM3XJBAT+MI3sh +P792X2WpSAJWoSmWMoqMz61yURs4WpKCaZjQhU3hOW +1tMYIzOGCGhFGwGUcPJQmZnF25T1pEOncOb/dnDQy +msLlJ/Vl+DgbPlylcP5bfUaN374O1hCk297Vso/dO +cuSWI3esju9dFdndPRx2uwzIhJLjrno7U/CkBbasH +rgbI7ts69BkcrjzJfagEsky9Fpm3Nc7ugzPbVg4bM +lDFLYLb3UV7zd7KwH/xd8xwOFslYVto0EO9XqPHKr +QP5bPUXaM/mbOYFrj9Lur2kWXr+5q5rEDtngURmk0 +l/IFQCqOhoLbVjqA68+Pnz59pmwewQS0dhSu1grU7 +gEruA1InPnlUQEyo3gB1bf0EOhTml1RHIz06mVkmT +E9IFZ12swbOryOTQ9HzbFKro6Nsq3R0Lm0AQ9f2/ +A6w7pBIW6wcJc41OGprDlLUgPnU6GOXo8MzaI0S5q +tmiq0Hit1GwYHTIMrUYasQmHjNaRqhaZCJkmtLEVX +536hoYxRGAsRjm9LWTwH6vnO6nhCLHQvYeBepXmL +2gRSsBrgFgD3FaEpbeq+q8piEI3z2W/pVTX7y0O9 +Zup/u0t3X3MP8Cutb6uvVN62bLag1ad1uPWietFy1 +345eNPzf+3ni7+fPm75t/bP6VQa9eyW2+alWuzbf/ +AOWNFyQ= +hc +b +�hc +L(1) +⇠q +A0 +ghc +ssoft +AOBnicrVdbc9tEFHZLgBINPDIi +4akMy3dGl3suHnwTJu205bJNGlLmzJR8KzktaXJ6tLVyq3Z2Xce+SW8AY/0gT/Bv+GsLra8coHOoLGd1Tnf+c5lzx4pXkrDjJvmXxcuvrfx/gcfXvpo8+NPtj797PL258+zJGc+eYnNGEvPJwRGsbkGQ85JS9SR +nDkUXLind9R+pMZYVmYxN/xeUrOIjyNw0noYw6i0fbGNdcj0zAWk2gyCSmRYkLm8ZThNJCbhrFUFqKv5VrYKBeQ10oAU5JRMuQguFDgptuRDzJBXcQtxBHKRXFmjsESp2XT8IR3d2JRi2lS9HGWi4U2oaCuH6m +Bq35Ui4KWGp4UvAJZY0AKhgMzEmfsK6WYBTMvRD5lOCShGkRsl4CLFXkPBHMnR4EPrnUswgiP9IQKUz5scl2QoCLE4e6BtYtodzw0gVUlULKCwo0wDyB8cSh/EFetayqJ16HxUlZ49O6erTp8+18cRf+HE2e5ty +ycBhwKjxIHJcr5yxyzcwPfgHakm40N86odViIREDybI05i1YtDq9tXOrSoEvQdi0BTV9L1MBOBX4SehWMyLwu7dW2oSL1NEh4EpdOyvUCZYMPu/DRbL4pjiIM9U/gYKhzo9yoVlL9e2VK84qsgk9HRhFp5VOCs +hqJjOohGrlCUpnmKeMC2rGt3dqxMvlkJYLcgUa1YhmCkdjOKbBGFXUWhM9tlFHUz1KWovflvKaTZVR2Eyv1STxuHPbN+r6YDKPLO2bXLC6jvbCqxU6nuo5H25f+dMeJn0ck5j7FWXZqmSk/E5jxEFoP6POMpN +g/x1NyCsYRyQ7E8VQk8YVkIyNScLgG3OjkDYtBI6ybB5gFQtn+k6JVynO835OaZCOM0hxr4paNJTg2eGpCGuOQEZ/TOSywz0KI1fADzLDPY6ueMkgqYCM5SZcbkxe+Qk0FlTMxTST6jcN8CiTmu7tKpIqHfy +GNIk1XcLGUpQT8EiulE5EOeUhS15JLQrA3ga+eiDclrKlf9TQP1qjP2roldtVwFEKIT3WzY7Sh5qVrm7qv2x3juJExatBAaH9S6hHBsHxtCwWhZRfu9E4fPRidQ1XqnwdBtMn759i+i9hc7l5DUXJNLTOIRn7OM +7d2W5GtdAJVp1ZbjJ4al1JhrzZgdSUE3jskE7ovWFmOYoVNGSCwFm0LUcLKQCeMC/so20qM5PKuf3D+QwnKQ+lh9DQZk8wVXD1Vfqbn17wLb/QJasO1bqPhoZH4DYvf6yO45yHZu6qiDZVgmhATfNtOhmj8NoG3 +1gGqA7L6tQ5/A4aqS3IdKIMvUa1GyLXF2H8hsWxGuyUAVtw5ubw/dXM/WBP4D3xHD8XSRhG2hQ/1emsZV6F9KJ+lvi3xzlLaY3bd5DTR5at72tJuQK1exZEaK4r4feEUhgNtWtb7Qhqe3749OkTbfMANmilo3CN +VrD2BlDJfUDqjo8f1BATqjdAjq2fQI/C/JLqaBRHpzQrhcUJWUX3ayByw5vY4vD0SJWybWxSbk1GrqSrsHD+zM8zrBuUIvXWPhzHOvwQrY+S9ICV1OhjV6ODM2iMUvW62r0HKstG0YHDANrkQlchUKG68hVSus +K2SWtcpSd3XFCw1luHEqRDy6LmV9H6n7G4v7mSfETGePI/UorR7QIpaA1wCpBriuHDaequr/oCgFIbz3WPpbTnvx3O5ae13ncW/n1kH1BnSp82Xnq87VjtUZdG51HnSO86/sbPG79t/LHxZunrV+2ft36vYRe +vFDZfNFZube/A3n4wQ+ +b +�hc +B1 +ghc +shc +Figure 3: +The SCET representations of the full theory diagram in Fig.1, see text. +The matching kernel K(ω, ω′, ω1, ω2) introduced in (44) can be computed perturbatively, i.e. to +extract K, we can replace B-meson state by a b-quark in the definition of the soft function and +calculate both sides of the equation on the partonic level. The LP soft function here appears +since the limit ω1,2 → ∞ is equivalent to the treatment of t and t′ as infinitesimal variables in +(32) and, consequently, the soft Wilson lines obtained from decoupling in the anti-hardcollinear +direction Sn cancel. At the same time, the softcollinear quark field produces an additional soft +Wilson line associated with the hardcollinear direction Sn because we require the softcollinear +quark to have the same gauge transformation as a hardcollinear field. Finally, the structure of +the soft function corresponds to LP shape function S(ω). Consistency of the second and third +refactorisation conditions, which approach the softcollinear limit from two different directions as +shown in Figure 2, leads to +� Λ +−p+ +dω �J (mb (p+ + ω) , u, u′) S(ω)� = +� Λ +−p+ +dωJg(mb(p+ + ω)) +� +dω′K(ω, ω′, ω1, ω2)S(ω′). +(45) +This relation implies that the kernel K can be obtained from the quark-gluon jet function in the +limit when momentum fraction of the quark tends to zero. Furthermore, it confirms that the +kernel K is a perturbative object and that the softcollinear scale can be treated perturbatively. +Finally, we note that softcollinear quarks must appear on both sides of the cut. Fermion num- +ber conservation implies that only in this case we get a non-vanishing decay rate. Consequently, +the endpoint divergences only appear in the limit when both u and u′ are small or when ω1 and +ω2 are large. +Figure 3 shows that the A- and B-type current have the same structure in the refactorisation +limit. On the left, the s-quark is soft and emitted through the insertion of the subleading power +Lagrangian. On the right, the s-quark is hardcollinear and emitted directly from the hard B-type +vertex. When the fraction of the hardcollinear momentum of the s-quark tends to zero, the B-type +current refactorises into the time-ordered product represented on the left, and both diagrams rep- +resent the same full theory configuration. This duality in the description leads to the appearance +of the endpoint divergences. A similar problem has already been identified in Refs. [49,50], in the +context of QED corrections in Bs → µ+µ− due to O7 operator at the amplitude level. +14 + +4.1 +Refactorisation at leading order +Based on the refactorisation conditions, we first discuss the procedure of refactorisation at the +leading order. We explicitly verify the conditions using the leading order results. Starting with the +last refactorisation condition, we consider the factorisation theorem of the A-type contribution +when the soft function is considered in the limit ω1 ∼ ω2 ≫ ω. This asymptotic limit of the +soft function can be analysed by means of semi-perturbative methods [51], where the energetic +softcollinear quarks are treated perturbatively, while ordinary soft modes are assumed to be +nonperturbative. In the leading order, this corresponds to the replacement of the softcollinear +quarks by a cut propagator. We anticipate the endpoint divergence in the convolution of the +soft and the anti-hardcollinear jet functions and use dimensional MS regularisation within the +calculation. We find the following expression of the asymptotic soft function at leading order [51],: +�S (ω, ω1, ω2) = CFA(ϵ) αs +(4π) ω1−ϵ +1 +δ(ω1 − ω2) +� Λ +ω +dω′ S(ω′) +�(ω′ − ω) +µ2 +�−ϵ +, +(46) +which includes the leading power shape function S(ω). Note that this expression, in principle, +receives corrections of higher order in αs and ΛQCD/ω1,2, which we do not take into account in +the leading order analysis within this section. A(ϵ) was defined in eq. (24) 9 +We convolute the asymptotic soft function with the anti-hardcollinear jet functions for large +ω1 and ω2 only by restricting the limits of the ω1 integral to mb and +∞. These integration +limits will become clear once we consider the B-type current contribution. Starting with the +factorisation formula of the A type current given in eq. (39), the asymptotic contribution of the +A type current reads at leading order: +dΓ +dEγ +|asy +A += 2N |CA0 +LO(mb)|2 +� Λ +−p+ +dωJLO +g +(mb(p+ + ω)) +� ∞ +mb +dω1JLO(ω1) +� ω1 +0 +dω2J +∗ +LO(ω2) �S(ω, ω1, ω2) += N|CA0 +LO (mb) |2 αsCF +(4π) mb +1 +ϵA(ϵ) +� Λ +−p+ +dω SLO(ω′) +�mb(ω + p+) +µ2 +�−ϵ +. +(47) +The 1 +ϵ pole is the manifestation of the endpoint divergence in the resolved contribution in the +limit ω1 ∼ ω2 ≫ ω. In the next step, we will see that the specific choice mb as a lower limit of +the ω1 integration is induced by the refactorisation conditions. The lower limit in the ω2 integral +can be chosen to be non-negative due to the delta function δ(ω1 − ω2). +Now we take the limit u → 0 in the factorisation theorem of the B-type current at leading +order, which we derived in eq. (25) before performing the integrals over u and u′. This leads to +dΓ +dEγ +|u,u′→0 +B += −N +��CA0 +LO (mb) +��2 +αsCF +(4π) mb +1 +ϵ A(ϵ) +� Λ +−p+ +dω SLO(ω) +�mb(ω + p+) +µ2 +�−ϵ +. +(48) +This result differs from eq. (47) only by an overall sign. The sum of these two terms is finite and +equal to zero. This leading-order result is a special case of the all-order relation, which follows +from refactorisation conditions. In the ω1 ∼ ω2 ≫ ω (asymptotic) limit of the A-type current +(with integration limits over ω1 from mb to +∞), we exactly single out the same term as in the +9We note that we do not confirm the leading order result of the asymptotic soft function of Ref. [4] in the +dimensional regularisation scheme. +15 + +u → 0 of the B-type current up to a minus sign. This reflects the fact that in the limits u → 0 +and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8 − O8 contribution have the same structure. +Moreover, we see that with the relations mbu = ω1 and mbu′ = ω2, the u, u′ → 1 limit corresponds +to the limit ω1, ω2 → mb, which fixes the integration limit in the subtraction term of the A-type +current. +We can summarise the relation we just verified at LO as +dΓ +dEγ +|asy +A += (−1) dΓ +dEγ +|u,u′→0 +B +. +(49) +The refactorisation conditions guarantee that the eq. (49) holds to all orders in perturbation +theory. To make this relation useful for the reshuffling of the factorisation theorem, let us consider +an integral of �S(ω1, ω2, ω)J(ω1)J +∗(ω2) over the ω1,2 ∈ [0, ∞]. Since �S(ω1, ω2, ω) is expanded for +ω1,2 ≫ ω, this integral is scaleless and equal to zero in dimensional regularisation. We then +perform the following manipulations of the integration limits +0 = +� ∞ +0 +dω1 +� ∞ +0 +dω2 = 2 +� ∞ +0 +dω1 +� ω1 +0 +dω2 = 2 +�� mb +0 +dω1 + +� ∞ +mb +dω1 +� � ω1 +0 +dω2. +(50) +In the second step, we made use of the fact that the integrand is symmetric in ω1 and ω2 as we +derived to all orders in eq. (38). Finally, we split the integration region into two parts suitable +for the subtraction. +The term integrated over ω1 from mb to ∞ is already in the form suitable for the subtraction +of the A-type term and equal to (47). To bring the second term into the form of eq. (48), we +perform substitutions ω1 = mb u and ω2 = mb u′, use (40) to replace J(ω1) by the singular part +of the CB1 matching coefficient and then use the second refactorisation condition to derive +2N +��CA0 +LO(mb) +��2 � mb +0 +dω1JLO(ω1) +� ω1 +0 +dω2J +∗ +LO(ω2) +� Λ +−p+ +dωJLO +g +(mb(p+ + ω)) �S(ω, ω1, ω2) += 2N +� 1 +0 +du +� +CB1 +LO (mb, u) +� � 1 +u +du′ � +CB1 +LO (mb, u′) +� � Λ +−p+ +dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� (51) +We rewrite Eq. (49) using the functions which enter the factorisation theorem: +2N +��CA0 +LO(mb) +��2 � ∞ +mb +dω1JLO(ω1) +� ω1 +0 +dω2J +∗ +LO(ω2) +� Λ +−p+ +dωJLO +g +(mb(p+ + ω)) �S(ω, ω1, ω2) += − 2N +� 1 +0 +du +� +CB1 +LO (mb, u) +� � 1 +u +du′ � +CB1 +LO (mb, u′) +� � Λ +−p+ +dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� . +(52) +We see with the help of (51) that the fact that the sum of asymptotic contributions is equal to +zero is a consequence of our refactorisation conditions. It is now clear that these two subtraction +terms, which add up to zero, make it possible to reshuffle the factorisation theorem and cancel +the endpoint divergences at the leading order. +16 + +4.2 +Bare refactorised factorisation theorem +The generalisation of the LO order result to all orders is straightforward. +Since we are still +working in d-dimensons with bare objects, we can insert a scaleless expression into the factorisation +theorem using the integral manipulations we performed at LO, see eq. (50) +Using the all-orders refactorisation conditions discussed at the beginning of this section, we +then can cast the subtraction term into the following form with the help of the same manipulations +as in the LO case and generalise eq. (52) to all orders: +0 = 2N +��CA0 (mb) +��2 � Λ +−p+ +dωJg (mb (p+ + ω)) +� ∞ +mb +dω1J (ω1) +� ω1 +0 +dω2J +∗ (ω2) �S (ω, ω1, ω2) ++ 2N +� 1 +0 +du +� +CB1 (mb, u′) +� � 1 +u +du′ � +CB1∗ (mb, u′) +� � Λ +−p+ +dω �J (mb (p+ + ω) , u, u′) S(ω)� . (53) +Starting from the all-order bare factorisation theorem +dΓ +dEγ += 2N +��CA0 (mb) +��2 � ∞ +−∞ +dω1J (ω1) +� ω1 +−∞ +dω2J +∗ (ω2) +� Λ +−p+ +dωJg (mb (p+ + ω)) S (ω, ω1, ω2) ++ 2N +� 1 +0 +duCB1 (mb, u) +� 1 +u +du′CB1∗ (mb, u′) +� Λ +−p+ +dωJ (mb (p+ + ω) , u, u′) S(ω) +(54) +and subtracting eq. (53) we arrive at +dΓ +dEγ +|A+B = 2N +� Λ +−p+ +dω +� +Jg(mb(p+ + ω)) +��CA0 (mb) +��2 +(55) +× +� ∞ +−∞ +dω1 +� ω1 +−∞ +dω2J(ω1) J +∗(ω2) +� +S (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �S(ω, ω1, ω2) +� ++ +� 1 +0 +du +� 1 +u +du′ � +CB1 +LO (mb, u) CB1∗ (mb, u′) J (mb (p+ + ω) , u, u′) S (ω) +− +� +CB1 (mb, u) +� � +CB1∗ (mb, u′) +� +�J (mb (p+ + ω) , u, u′) S(ω)� +�� +, +where �J (mb (p+ + ω) , u, u′) S(ω)� = Jg(mb(p+ + ω)) �S(ω, mbu, mbu′) and +� +CB1 (mb, u′) +� += (−1)CA0 (mb) mbJ (umb). We note here that the second term effectively restricts the integration +range over ω1 to a finite range in the first line and consequently removes endpoint divergence. +Thus these terms need to be added together before the ω1 integral is performed. Similarly, the +last term removes the endpoint divergence of the third term, and therefore u integration has to be +performed after these two terms are added up. In addition, we note that the integrals in the first +term are finite for large negative values of ω1 and ω2 due to nonperturbative dynamics [4]. At this +point, the convolutions integrals in the A- and B-type contributions are no longer divergent, and +we can renormalise the functions entering the factorisation theorem and take the limit d → 4. +4.3 +Refactorised factorisation theorem after renormalisation +We achieved refactorisation at the level of the bare factorisation theorem. It has been pointed +out that refactorisation and renormalisation do not commute in general [23, 52]. Therefore, for +17 + +the result to be helpful for the resummation of the large logarithms, we must prove that we +can express the factorisation theorem in terms of renormalised objects. To this end, we have to +replace bare quantities with renormalised ones. The renormalisation of hard matching coefficients +is well-established +CA0 +bare(mb) = ZA0(µ) CA0 +ren(µ, mb) , +(56) +CB1 +bare(u) = +� 1 +0 +du′ ZB1(µ, u, u′) CB1 +ren(µ.u′) , +(57) +where the one-loop renormalisation factors can be found in Ref. [43]. The LP jet function is +renormalised according to +Jbare +g +(p2) = +� p2 +o +dp′2 ZJg(µ, p2 − p′2) Jren +g (µ, p′2) , +(58) +with the ZJg factor given in Refs. [53, 54] up to the three-loop order. Similarly, the LP shape +function +Sbare(ω) = +� +dω′ ZS(µ, ω − ω′) Sren(µ, ω′) +(59) +is well-known [55] +Much less is known about NLP objects. The radiative jet function is a notable example which +appeared before in the context of B → γℓν [12]. It has recently been computed at the two-loop +order in Ref. [17]. The most important detail is that the time-like (ω > 0) and space-like (ω < 0) +radiative jet functions do not mix under renormalisation +J ++ +bare/ren(ω) = θ(ω)Jbare/ren(ω) , +(60) +J +− +bare/ren(ω) = θ(−ω)Jbare/ren(ω) , +(61) +and +J ++ +bare(ω) = +� ∞ +0 +dω′ Z+ +J (µ, ω, ω′) , J ++ +ren(µ, ω′) , +(62) +J +− +bare(ω) = +� 0 +−∞ +dω′ Z− +J (µ, ω, ω′) J +− +ren(µ, ω′) . +(63) +This separation into time-like and spec-like jet functions is necessary since we choose to integrate +the subtraction term only over non-negative values of ω1,2. Finally, we define the renormalisation +of the NLP soft and jet functions +Sbare(ω, ω1, ω2) = +� +dω′dω′ +1dω′ +2 ZS(µ, ω, ω′, ω1, ω′ +1, ω2, ω′ +2) Sren(µ, ω′, ω′ +1, ω′ +2) , +(64) +Jbare(p2, u1, u2) = +� +dp′2 +� 1 +0 +du′ +1 +� 1 +0 +du′ +2 ZJ(µ, p2 − p′2, u1, u′ +1, u2, u′ +2) Jren(p′2, u′ +1, u′ +2) . +(65) +These renormalisation kernels are currently unknown. +18 + +We require that A- and B-type contributions are separately RG invariant (see Ref. [56] for +analogous treatment). This leads to the following conditions on the renormalisation kernels +|ZA0|2 +� +dω +� +dω1 +� +dω2 ZJg(ω − ω′)ZJ(ω1, ω′ +1) Z† +J(ω2, ω′ +2) ZS(ω, ω′′, ω1, ω′′ +1, ω2, ω′′ +2) +=δ(ω′ − ω′′) δ(ω′ +1 − ω′′ +1) δ(ω′ +2 − ω′′ +2) , +(66) +and +� 1 +0 +du1 +� 1 +0 +du2 +� +dω ZB1(u1, u′ +1) ZB1†(u2, u′ +2) ZJ(ω − ω′, u1, u′ +1, u2, u′ +2) ZS(ω − ω′′) += δ(ω′ − ω′′) δ(u′ +1 − u′′ +1) δ(u′ +2 − u′′ +2) ; +(67) +and further, RG invariance of the subtraction term leads to +|ZA0|2 +� ∞ +0 +dω1 +� ∞ +0 +dω2 +� +dω Z+ +J (ω1, ω′ +1) Z+† +J (ω2, ω′ +2) ZJg(ω − ω′) Z�S(ω − ω′′, ω1, ω′′ +1, ω2, ω′′ +2) += δ(ω′ − ω′′) δ(ω′ +1 − ω′′ +1) δ(ω′ +2 − ω′′ +2) . +(68) +These conditions are sufficient to prove that renormalisation and refactorisation commute and +there is no leftover term. +We can now insert the above definitions into eq. (55), +dΓ +dEγ +|A+B = 2N +� Λ +−p+ +dω +� +Jren +g (mb(p+ + ω)) +��CA0 +ren (mb) +��2 +(69) +× +� ∞ +−∞ +dω1 +� ω1 +−∞ +dω2J ++ +ren(ω1) J ++∗ +ren(ω2) +� +Sren (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �Sren(ω, ω1, ω2) +� ++ +� 1 +0 +du +� 1 +u +du′ � +CB1 +ren (mb, u) CB1∗ +ren (mb, u′) Jren (mb (p+ + ω) , u, u′) Sren (ω) +− +� +CB1 +ren (mb, u) +� � +CB1∗ +ren (mb, u′) +� +�Jren (mb (p+ + ω) , u, u′) Sren(ω)� +�� +. +This is our final result. Endpoint divergences are manifestly absent, assuming one performs +the integrals over ω1 after adding the first and second terms together. Similarly, the integrals +over u should be performed after adding the last two lines. +This renormalised factorisation +theorem allows for a consistent resummation of large logarithms within the resolved O8 − O8, +using standard RG methods owing to the fact that each object appearing in the above equation +is a single scale object. However, a judicious choice of scale might be necessary. +19 + +5 +Summary and Outlook +In the present paper, we identified the divergences in the resolved, but also in the direct subleading +O8 − O8 as endpoint divergences which lead to a breakdown of the factorisation theorem already +at leading order in four space-time dimensions. The failure of naive factorisation does not allow +for consistent separation of scales and, consequently, resummation of large logarithms. +However, it was recently shown [9] that the resolved contributions still represent the most +significant uncertainty in the inclusive ¯B → Xsγ decay. Large scale dependence and also a large +charm mass dependence were identified in the lowest order result of the resolved contribution, +which calls for a systematic calculation of αs corrections and RG summation of all resolved +contributions [9]. A mandatory input for this task is a well-defined factorisation formula for these +subleading corrections. This critical step was established in this paper. The next step consists of +computing renormalisation kernels for the NLP soft and jet functions, extracting the anomalous +dimensions and solving the RG equations to resum large logarithms. +Recent intensive studies of the power corrections in collider applications of SCET [19,22–24, +42,52] lead to the development of new techniques that allow for a reshuffling of terms within the +factorisation formula so that all endpoint divergences cancel out. We used these new techniques in +our flavour application which includes nonperturbative functions typically not present in collider +applications of SCET. Unlike in the h → γγ decay [22], in the considered SCETI problem, there +are no leftover terms present after renormalisation. +To derive a consistent factorisation theorem, we first established the bare factorisation theorem +for the resolved and direct contributions on the operatorial level. Then we derived the all-orders +refactorisation conditions applicable to our process. This idea is based on the fact that in certain +limits, the two terms of the subleading O8 − O8 contribution have the same structure, which +guarantees that the endpoint divergences cancel between the two terms to all orders. Finally, we +proved that we could express the factorisation theorem in terms of renormalised objects so that the +result can be used for the resummation of the large logarithms within the resolved contributions. +Acknowledgements +We thank Martin Beneke and Matthias Neubert for their valuable discussions. RS would also like +to thank Mathias Garny and Jian Wang for many discussions on power corrections in SCET and +Mikolaj Misiak for a discussion on the theoretical predictions for B → Xsγ. TH is grateful to +Michael Benzke for uncounted discussions on the resolved contributions. RS is supported by the +United States Department of Energy under Grant Contract DESC0012704. 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Fleming, “Renormalization and Scale +Evolution of the Soft-Quark Soft Function,” JHEP 07, 104 (2020) [arXiv:2005.03013 [hep- +ph]]. +24 + diff --git a/G9AzT4oBgHgl3EQfxf6F/content/tmp_files/load_file.txt b/G9AzT4oBgHgl3EQfxf6F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5183db2abc553fdf2a277de0b2a91bae6fd8f6eb --- /dev/null +++ b/G9AzT4oBgHgl3EQfxf6F/content/tmp_files/load_file.txt @@ -0,0 +1,1046 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf,len=1045 +page_content='MITP/21-047 Refactorisation in subleading ¯B → Xsγ Tobias Hurtha and Robert Szafron,b aPRISMA+ Cluster of Excellence and Institute of Physics (THEP), Johannes Gutenberg University, D-55099 Mainz, Germany bDepartment of Physics, Brookhaven National Laboratory, Upton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=', 11973, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Abstract We establish refactorisation conditions between the subleading O8-O8 contributions to the inclusive ¯B → Xsγ decay suffering from endpoint divergences and prove a factorisa- tion theorem for these contributions to all orders in the strong coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This allows for higher-order calculations of the resolved contributions and consistent summation of large logarithms, consequently reducing the recently found large-scale dependence in these contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We implement the concept of refactorisation in a heavy flavour application of SCET, which includes nonperturbative functions as additional subtlety not present in collider applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='01739v1 [hep-ph] 4 Jan 2023 Contents 1 Introduction 3 2 General setup 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1 Hard matching .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 10 4 Refactorisation of the endpoint contribution 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1 Refactorisation at leading order .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='2 Bare refactorised factorisation theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='3 Refactorised factorisation theorem after renormalisation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 17 5 Summary and Outlook 20 2 1 Introduction There has been a general belief that soft-collinear factorisation at subleading power in ΛQCD/mb expansion is well established for inclusive B−decay modes such as ¯B → Xsγ, ¯B → Xsℓℓ, or ¯B → Xuℓ¯ν [1] – in contrast to exclusive B decays where factorisation theorems do not exist at the subleading power in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' There are two types of subleading contributions to the inclusive ¯B → Xsγ decay, direct and resolved ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the latter, the photon does not directly couple to an effective electroweak vertex, but they contain subprocesses in which the photons couple to light partons instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' These subleading corrections are nonlocal in the endpoint region, and they stay nonlocal even in the region where the local heavy mass expansion is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In this sense, they represent an irreducible uncertainty of this decay mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Analogous subleading contributions exist in the inclusive ¯B → Xsℓℓ decay but not in the inclusive ¯B → Xuℓ¯ν decay because, in this case, the leptons can couple to light partons via the W vector boson only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The first systematic analysis of resolved contributions to the inclusive ¯B → Xsγ decay [2,3] was worked out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4,5], the corresponding 1/mb contributions to the inclusive ¯B → Xsℓℓ decay were discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [6,7], using soft collinear effective theory (SCET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Recently, the uncertainty due to the resolved contribution was reduced with the help of a new hadronic input [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' But these resolved contributions still represent the largest uncertainty in the inclusive ¯B → Xsγ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Moreover, a large scale dependence and also a large charm mass dependence were identified in the lowest order result of the resolved contribution, which calls for a systematic calculation of αs corrections and renormalisation group (RG) summation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' A mandatory prerequisite for this task is an all-order in the strong coupling constant αs factorisation formula for the subleading power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The factorisation of resolved contributions introduces a new ingredient, namely an anti- hardcollinear jet function [4], typically referred to as a radiative or amplitude-level jet function in collider and flavour applications [10–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' They are not represented by cut propagators as the usual jet functions but as full propagator functions (both dressed by Wilson lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' But as al- ready noticed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4], the specific resolved O8 − O8 contribution does not factorise because the convolution integral is UV divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4] emphasised that there is an essential difference between divergent convolution integrals in power-suppressed contributions of exclusive B decays and the divergent convolution integrals in the present case, while the former were of IR origin, the latter divergence of UV nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' However, a solution at the lowest order was established by considering the sum of direct and resolved O8 − O8 contributions, which was shown to be scale and scheme dependent by using a hard cut-off in the resolved contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' But the failure of factorisation does not allow for a consistent resummation of large logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In this paper, we identify the divergences in the resolved and in the direct contributions as endpoint divergences by showing that also the divergence in the direct contribution can be traced back to a divergent convolution integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Recently, new techniques were presented in specific collider applications of SCET [19–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The so-called refactorisation conditions or endpoint factorisation [22,23,25–27] allow for an operator-level reshuffling of terms within the factorisation formula so that all endpoint divergences cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In this work, we now implement this idea in a flavour application of SCETI, which includes nonperturbative soft functions not present in collider applications – often referred to as subleading shape functions [28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' As a first step, we derive the matching of the hard function for the two operators involved in the O8 − O8 subleading contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the second step, we establish the bare factorisation theorem for the direct and the resolved contribution at an operational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Then we derive 3 the refactorisation conditions to all orders, leading us finally to the renormalised factorisation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We present all steps for the inclusive ¯B → Xsγ, but all the details can also be taken over for the corresponding ¯B → Xsℓℓ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 2 General setup The starting point for all calculations concerning the ¯B → Xsγ decay is the weak effective Lagrangian defined at a scale µb parametrically equal to the b-quark mass µb ∼ mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The weak effective Lagrangian is obtained from the SM Lagrangian after integrating out the heavy particles like the heavy gauge bosons and the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We use the convention of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Assuming Standard Model CKM unitarity, with λq = VqbV ∗ qs and λu + λc + λt = 0, the effective Hamiltonian may be written as Heff = GF √ 2 � q=u,c λq � C1 Oq 1 + C2 Oq 2 + C7γ O7γ + C8g O8g + � i=3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=',6 Ci Oi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (1) Here we concentrate on the O8 operator: O8g = − gs 8π2 mb ¯sσµν(1 + γ5)Gµνb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (2) Our sign convention is that iDµ = i∂µ+gs taAa µ+e qfAµ, where ta are the SU(3) colour generators, and Qf is the electric charge of the fermion in units of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We consider the CP-averaged ¯B → Xsγ photon energy spectrum in the endpoint region where Mb − 2Eγ = O(ΛQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Soft-collinear- effective theory (SCET) offers the appropriate framework for this multi-scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The kinematics of the decay is given as follows: the initial meson carries momentum pB, and it decays into a photon with momentum q and a jet whose total momentum is PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' From pB − q = PX in the B meson rest frame, we have 2MBEγ = M 2 B − M 2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Thus, the jet invariant mass MX is much smaller than the photon energy Eγ and jet energy EX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We set PX⊥ = 0 and choose reference vectors n2 = n2 = 0, v2 = 1, such that n + ¯n = 2v and nn = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Choosing qµ = Eγ¯nµ and pµ B = MBvµ , (3) we find MB = ¯nPX and MB = nPX + 2Eγ or equivalently P µ X = (MB − 2Eγ)nµ + MB¯nµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (4) Thus, there is only one independent kinematical variable in the ¯B → Xsγ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' One may choose the photon energy Eγ or nPX = MB − 2Eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Three dynamical scales describe the endpoint region, a hard scale of O(MB), an intermedi- ate (anti-)hardcollinear scale of O( � MBΛQCD), and a soft scale of O(ΛQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The expansion parameter in our present analysis is defined as λ2 = ΛQCD/MB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The photon can be treated as anti-hardcollinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The hadronic final state factorises into a hardcollinear jet and soft wide-angle radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Since the soft modes have parametrically smaller virtuality than the hardcollinear 1Alternative convention often used in the literature is to define λ = ΛQCD/MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 4 modes, the problem at hand is described by the SCETI setup [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Using a shorthand nota- tion a ∼ (na, ¯na, a⊥) to indicate the scaling of the momentum components in powers of λ, we have: hard momentum scales like phard ∼ (1, 1, 1)mb, a hardcollinear one phc ∼ (λ2, 1, λ)mb, an anti-hardcollinear region phc ∼ (1, λ2, λ)mb and a soft momentum psoft ∼ (λ2, λ2, λ2)mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The first step in the derivation of a factorisation theorem is hard matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We have to match the electroweak operator onto SCET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We will see that the direct contribution is represented by a next-to-leading power (NLP) B-type current in SCET, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' power-suppressed current composed of two collinear building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The resolved contribution is represented by a time-ordered product of a leading-power (LP) A-type current with a subleading L(1) ξq Lagrangian (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [19, 35–37] for a precise definition of the different types of currents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the second step, we integrate out the hardcollinear fields, which lead to the appearance of jet functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The latter are, technically speaking, matching coefficients of SCET on the pure soft effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Kinematics forbids the emission of anti-hardcollinear partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Thus anti-hardcollinear fields have to be integrated out at the amplitude level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The physics in the anti-hardcollinear direction is similar to the threshold Drell-Yan expansion, where kinematical constraints forbid hardcollinear emissions into the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' For more details on SCET NLP factorisation and resummation in the threshold Drell-Yan, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [13,14,20,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the hard- collinear sector, the formalism resembles, for example, the thrust factorisation, where NLP jet functions are defined at the cross-section level [15,39–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1 Hard matching The electroweak operator O8 matches onto two possible SCET operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' First, we consider the A-type current, which enters the resolved contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Within the present section, the SCET operators are always given before the soft decoupling transformation [32] is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The A-type SCET operator is given by OA0 8g (0) = χhc (0) /n 2γµ⊥Aµ hc⊥ (0) (1 + γ5) h (0) , (5) where h is the heavy quark field, and the SCET building blocks are hardcollinear gauge-invariant due to the introduction of hardcollinear Wilson lines W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The fermionic building block is χhc = W † hcξ and gluon field is Aµ hc⊥ = W † hc [Dµ hc⊥Whc] = Aaµ hc⊥ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (6) Note that the colour and Dirac structure for the A-type operator is uniquely fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' For the matching, we can use the partonic QCD amplitude b (pb) → s (ps) g (r), where the momenta pb is hard, ps anti-hardcollinear and r hardcollinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The matching condition is given by GF √ 2λt C8g ⟨Q8g⟩ = CA0 (mb) � OA0 8g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (7) The brackets ⟨ ⟩ indicate that the matrix element of the operators is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At leading order in αs we find CA0 LO (mb) = m2 b 4π2 GF √ 2λqC8g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 5 The B-type current which enters the direct contribution is the following SCET operator:2 OB1 8g (u) = � dt 2πe−iumbtχhc (t¯n) γν⊥ Qs Bν hc⊥ (0) γµ⊥ Aµ hc⊥ (0) (1 + γ5) h (0) , (8) with Qs as the electric charge of the strange quark in units of e and the electromagnetic gauge- invariant transverse photon field Bν hc⊥ = e � Aν ⊥ − ∂ν ⊥ n∂ nA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (9) We note that beyond the LO, a second operator with an alternative Dirac structure γµγν appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' These operators do not mix under renormalisation [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We checked by explicit computation at the one-loop order that the matching coefficient for the operator with γµγν structure does not develop any endpoint divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We use the partonic QCD amplitude b (pb) → g (r) s (ps) γ (q) to fix the matching coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Here the momenta are pb hard, r and ps hardcollinear and q anti-hardcollinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' On the QCD side, a time-ordered product of the operator O8 and of the QED current, LQED,q (x) = eq Aµ (x) q (x) γµq (x) , (10) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The matching condition at the leading order in QED is given by GF √ 2λtC8g i � d4x ⟨T [O8g (0) , LQED,b (x) + LQED,s (x)] ⟩ = � 1 0 duCB1 (mb, u) � OB1 8g (u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (11) We do not consider QED corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At leading order in αs, we find that only the QED current with an s quark contributes, and we arrive at CB1 LO (mb, u) = (−1)u u m2 b 4π2 GF √ 2λt C8g = (−1)u uCA0 LO (mb) , (12) where we use hardcollinear momentum conservation ¯ns + ¯nr = mb and introduce hardcollinear momentum fraction u = ¯nps mb and u = 1 − u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we compare the different kinematics of the A- and B-type currents in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The external s-quark carries hardcollinear momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Therefore the intermediate propagator is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This situation is represented in SCET by the B-type current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' When the momentum of the external s-quark tends to zero, the propagator becomes anti-hardcollinear and cannot be integrated out – it must be reproduced by a dynamical field in the low energy EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This situation is represented in SCET by the time-order product of subleading Lagrangian and the A-type current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The degeneracy in the EFT description is the reason why the SCET develops divergencies in the convolution integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 2Note that this operator is equal to J2 (τ) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [43] (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 16), with J2 (τ) = 2mbOB1 8g (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='AOBHicrVdbc9tEFHZLgGItPDIi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='4YkQ0u3RpLtOHnwTJu205bJNGlLmzJR8KzktaXJ6pLVyq3Z2Vce+S ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='O8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='ghc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='shc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='Figure 1: The full theory LO diagram which induces an endpoint divergence in SCET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 3 Bare factorisation theorem The derivation of the factorisation theorem follows the standard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We first perform the soft decoupling transformation [32], but we do not use a new notation for the hard- collinear fields after decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The decay rate is obtained from the imaginary part of the current-current correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The states factorise and thus allow taking matrix elements separately in hardcollinear, anti-hardcollinear and soft sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The common to both A- and B-type contributions is the anticollinear matrix element of the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' It is given by a discontinuity of the photon propagator −gµν ⊥ e2 Jγ � q2� = 1 2πi Disc[ i � d4xeiqx ⟨0| T [Bµ hc⊥ (x) , Bν hc⊥ (0)] |0⟩ ] (13) = −gµν ⊥ e2 δ � q2� θ � q0� = −gµν ⊥ e2 δ+ � p2� (14) Since we are only interested in the photon-final state, the above expression is exact to all orders in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1 B-type current (direct) contribution There are several functions entering the factorisation formula of the direct contribution with the B-type current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The soft function – the leading power shape function – is defined as S (ω) = 1 2mB � dt 2πe−iωt ⟨B| h (tn) Sn (tn) S† n (0) h (0) |B⟩ , (15) or with open indices [46,47]3 1 2mB � dt 2πe−iωt ⟨B| [hα (tn) Sn (tn)]i � S† n (0) hβ (0) � j |B⟩ = δij 2 Nc �1 + /v 2 � βα S (ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (16) 3We use Greek for spinor indices and Latin for colour indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 7 The hardcollinear jet function is a genuine NLP object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In analogy to the LP jet function, we define it as a vacuum matrix element of a product of hardcollinear fields J � p2, u, u′� = (−1) 2Nc 1 2π � dtdt′ (2π)2 d4x e−imb(ut−u′t′)+ipx (17) Disc � ⟨0| tr �1 + /v 2 (1 − γ5) /Ahc⊥ (x) γν ⊥χhc (t′¯n + x) χhc (t¯n) γν⊥ /Ahc⊥ (0) (1 + γ5) � |0]⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The field operators are time- or anti-time-ordered according to the Keldysh formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='4 The trace is taken both with respect to colour and spinor spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Using projection properties of the hardcollinear fields and 2/v = /n+ + /n− the Dirac algebra in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (17) can be simplified to J � p2, u, u′� = (−1) 2Nc 1 2π � dtdt′ (2π)2 d4x e−imb(ut−u′t′)+ipx (d − 2)2 (18) Disc � ⟨0| tr �/¯n 4(1 − γ5)Aµ hc⊥ (x) χhc (t′¯n + x) χhc (t¯n) Ahc⊥ µ (0) (1 + γ5) � |0⟩ � where, as mentioned before, the trace is also applied in the colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' With these definitions, we find the bare factorisation theorem for the direct contribution dΓ dEγ = NB � 1 0 duCB1 (mb, u) � 1 0 du′CB1∗ (mb, u′) � Λ −p+ dωJ (MB (p+ + ω) , u, u′) S (ω) (19) with the prefactor NB = [(2π)] [e2Q2 s] [ 1 (2π)3 2 Eγ E2 γ 4π] = e2Q2 s Eγ 2π (20) The three pieces of the prefactor correspond to the phase space factors of the photon, to its charges and to the redefinition of the jet function with a 2π factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we prove to all orders in αs that the jet-function is symmetric in u and u′ up to complex conjugation: J(p2, u, u′) = J∗(p2, u′, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (21) This can be read off from the factorisation theorem of the direct contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The photon energy spectrum is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The leading power shape function is also real to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This can be shown by complex conjugation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (15) and by using translation invariance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Then the jet function inherits the symmetry property given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (21), from the product of the Wilson coefficients, CB1 (mb, u) CB1∗ (mb, u′), in the convolution integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' An anti-symmetric part of the jet function would cancel out in the convolution integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We emphasise that this property is also valid when the other B-type operator with the reversed Dirac structure is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In particular, the sum of the two mixed terms has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the latter terms, the reduction of the Dirac structure leads to (4 − d) (d − 2), and hence these terms vanish for d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The symmetry property is crucial for the refactorisation because it implies that no double subtraction regarding the variables u and u′ is needed in the B-type (direct) current contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This can be seen in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We showed above that the integrand of the convolution integral of the Wilson coefficients and the jet function in the two variables u and ¯u is real, so the 4For a brief summary, see appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 8 complete integrand is symmetric in u and u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Then the subsequent rearrangement is possible (we here only write the convolution variables u and u′): � 1 0 duCB1 (u) � 1 0 du′CB1∗ (u′) J (u, u′) = 2 � 1 0 duCB1 (u) � 1 u du′CB1∗ (u′) J (u, u′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (22) As the endpoint divergence manifests for small u and u′, we need to ensure that only the last integral over u is rendered finite by an appropriate subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At the leading order, the jet function is real, and we find that the jet function is symmetric in u and u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Explicitly, we find using the dimensional MS regulator (µ2ϵ → µ2ϵ exp(γEϵ)/(4π)ϵ): J � p2, u, u′� = CF αs 4π mb θ(p2) A(ϵ) δ(u − u′)u1−ϵ(1 − u)−ϵ �p2 µ2 �−ϵ , (23) with A(ϵ) = (2 − 2ϵ)2 (1 − 1/2 ϵ) Γ(1 − ϵ)−1 exp(γEϵ) = 4 − 10ϵ + O(ϵ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (24) We compute the convolution integrals explicitly5 using this leading order result for the jet function and also the hard function at leading order, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (12), dΓ dEγ |B = 2NB ��CA0 LO (mb) ��2 � 1 0 du ¯u u � 1 u du′ ¯u′ u′ (25) CFA(ϵ) αs (4π) mb � Λ −p+ dω S (ω) �mb(p+ + ω) µ2 �−ϵ u1−ϵ(1 − u)−ϵδ(u − u′) = NB ��CA0 LO (mb) ��2 CF αs (4π) mb � Λ −p+ dω S(ω) A(ϵ)B(3 − ϵ, −ϵ) �mb(ω + p+) µ2 �−ϵ , where B(x, y) denotes the Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We see that the divergence in the direct contribution is now identified as an endpoint point divergence in the convolution integral of the hard and the jet function in the u integration for u ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We emphasise that this endpoint divergence can be regularised within the dimensional regu- larisation scheme6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This leads to additional poles after performing the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Consequently, due to endpoint divergences, the bare factorisation formula is already invalid for the d → 4 limit at the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 5Symmetry of the original integral implies that � 1 u du′δ(u − u′) = θ(0) with θ(0) = 1/2, for u ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 6We note that we do not confirm the leading order result of the direct contribution of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4] in the dimensional regularisation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4] we get F (a) 88 (Eγ, µ) = CF αs(µ) 4π � mb 2Eγ �2 � ¯Λ −p+ dω �2 9 ln mb(ω + p+) µ2 + 2 9 � S(ω, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='2 A-type (resolved) contribution For the resolved contribution with the A-type current, we start with the time-ordered product OTξq = i � ddxT � Lξq (x) , OA0 8g (0) � = i � ddxT � qs (x+) Sn(x+) � Qs /Bhc⊥ + /Ahc⊥ � (x) χhc (x) , χhc (0) S† n(0)Sn(0)/n 2 /Ahc⊥(0) (1 + γ5) S† n(0)h (0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (26) The operator in the hardcollinear sector contains only gluon fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Hence the standard leading power gluon jet function appears −g2 sδabgµν ⊥ Jg � p2� = 1 2πi Disc � i � d4xeipx ⟨0| T � Aaµ hc⊥ (x) , Abν hc⊥ (0) � |0⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (27) At leading order we find the standard result Jg (p2) = δ+ (p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Besides photons, there are no energetic particles emitted in the anti-hardcollinear directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Thus, the anti-hardcollinear jet function is defined at the amplitude level: OTξq = � dω � dt 2πe−itω [qs]α (tn) � J (ω) �a νµ αβ Qs Bν hc⊥ (0) Aµ hc⊥ (0) [h (0)]β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (28) The anti-hardcollinear jet function can be decomposed as � J (ω) �a νµ αβ = J (ω) ta � γν ⊥γµ ⊥ /¯n/n 4 � αβ , (29) to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The other structure γµ ⊥γν ⊥ does not appear as one can read off from the structure of the T product in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (26) and the fact that the gluon and heavy quark fields are only spectators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The Dirac structure can then be simplified at the level of the cross-section with the help of the following relation: � γν ⊥γµ ⊥ /¯n/n 4 � αβ � γµ ⊥γν ⊥ /n/¯n 4 � α′β′ = (d − 2)2 �/¯n/n 4 � αβ �/n/¯n 4 � α′β′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (30) At leading order, the anti-hardcollinear jet function is given by � J (ω) �a νµ αβ = ta (ω + i ϵ) � γν ⊥γµ ⊥ /¯n/n 4 � αβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (31) Having defined hardcollinear and anti-hardcollinear functions, we now focus on the soft sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The operatorial definition of the soft function in position space with open Dirac indices is Sαβ,α′β′ (u, t, t′) = = g2 s ⟨B| � h (un) (1 − γ5) � α′ � Sn (un) taS† n (un) � S¯n (un) � S† ¯n (t′¯n + un) qs (t′¯n + un) � β′ × [qs (t¯n) S¯n (t¯n)]α S† ¯n (0) � Sn (0) taS† n (0) � [(1 + γ5)h (0)]β |B⟩ / (2mB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (32) 10 We can now plug in all the objects into the matrix element squared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' and we find the resolved contribution dΓ dEγ = NA ��CA0 (mb) ��2 � Λ −p+ dωJg (mb (p+ + ω)) � dω1 � dω2J (ω1) J ∗ (ω2) S (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' ω2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (33) with the prefactor NA = NB ≡ N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (34) and the scalar soft function obtained after contracting spinor indices according to S (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' t′) = (d − 2)2g2 s ⟨B| h (un) (1 − γ5) � Sn (un) taS† n (un) � S¯n (un) S† ¯n (t′¯n + un) (35) /n/¯n 4 qs (t′¯n + un) qs (t¯n) /¯n/n 4 S¯n (t¯n) S† ¯n (0) � Sn (0) taS† n (0) � (1 + γ5) h (0) |B⟩ / (2mB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The soft function in momentum space, which appears in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (33), is obtained through the Fourier transform of the position space expression according to S (ω, ω1, ω2) = � du 2πe−iuω � dt 2πe−itω1 � dt′ 2πeit′ω2S (u, t, t′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (36) As the NLP jet function in u and u′ variables, the soft function S (ω, ω1, ω2) is symmetric in ω1 and ω2 up to complex conjugation: S (ω, ω1, ω2) = S∗ (ω, ω2, ω2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (37) This property stems from the fact that the gluon jet function is real to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Thus, the soft function inherits the symmetry property from the product of the anti-hardcollinear jet func- tions, J (ω1) J ∗ (ω2) in the factorisation formula of the resolved contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Any anti-symmetric part would cancel in the convolution integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This symmetry property implies that within the refactorisation, a double-subtraction regarding the variables ω1 and ω2 in the A-type current contribution is not needed either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The symmetry implies that the integrand in the convolution integral between anti-hardcollinear jet functions and soft function in the two variables ω1 and ω2 is real and symmetric and allows for the following rearrangement of the convolution integral � ∞ −∞ dω1 � ∞ −∞ dω2J (ω1) J ∗ (ω2) S (ω1, ω2) = 2 � ∞ −∞ dω1 � ω1 −∞ dω2J (ω1) J ∗ (ω2) S (ω1, ω2) , (38) which is motivated by the fact in the resolved contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' As we will see explicitly in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1, the convolution integral of the jet and shape function is logarithmically divergent for ω1,2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At leading order, we find the factorisation formula in case of the A-type current (resolved) contribution:7 dΓ dEγ = 2N ��CA0 LO (mb) ��2 � Λ −p+ dωδ (mb (p+ + ω)) � ∞ −∞ dω1 � ω1 −∞ dω2 1 (ω1 − iϵ) 1 (ω2 + iϵ) S (ω, ω1, ω2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (39) We keep the soft function unevaluated at this point since this is a nonperturbative object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' For ω1,2 ≫ ω, the soft function can be shown to be asymptotically constant, which leads to endpoint divergence in the convolution integrals for large ω1,2 (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 7This confirms the leading order result of the resolved contribution of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4] when the asymptotic limit of the soft function is not yet considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 11 Figure 2: Scales relevant to refactorisation of the endpoint divergent contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The left part of the diagram represents the standard hierarchy of three scales for SCETI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Near the endpoint, when the momentum fraction u is no longer u ∼ O(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' u ≪ 1, we introduce additional, unphysical scales which make it possible to factorise further objects appearing in the bare factorisation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 4 Refactorisation of the endpoint contribution We here state the three refactorisation relations, which are based on the fact that in the limits u ∼ u′ ≪ 1 and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8−O8 contribution have the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The refactorisation relations are operatorial relations that guarantee the cancellation of endpoint divergences between the two terms to all orders in αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The refactorisation conditions result from the overlap between soft and hardcollinear modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The hierarchy of scales near the endpoint is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We will refer to these overlap modes as softcollinear modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' They play a similar role as the z-SCET modes introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [24] to prove the refactorisation of the B1-type matching coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The parameter z corresponds to the momentum fraction u in the present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' On the one hand, we can think of the softcollinear mode as a limit of hardcollinear mode when the large momentum fraction tends to zero8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' On the other hand, the softcollinear modes can be understood as a limit of the soft modes when the n+k momentum component becomes much larger than the remaining components, mb ≫ n+k ≫ λ2mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We want to emphasise that softcollinear and u-hardcollinear modes are not physical but help introduce refactorisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The softcollinear fields obey the same projection properties and have the same transformation properties regarding gauge invariance as their hardcollinear counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [22–24], we find that in the limit u → 0, the matching coefficient can be 8Thus, they do not appear in the leading power problems, where only operators with a single hardcollinear field in each direction occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 12 2 hard B1 CAO 7 umb u-hardcollinear hardcollinear J&S→J,3 softcollinear S↑ softfurther factorised � CB1 (mb, u) � = (−1)CA0 (mb) mbJ (umb) , (40) where �g(u)� only denotes the leading term of a function g(u) in the limit u → 0 and without any higher power corrections in u ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The function J (umb), which appears here, is exactly the same radiative jet function (29) we introduced before in the context of A-type contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This refactorisation condition stems from the fact that in the limit u → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' the amplitude used in the matching of the B-type current can be represented by a time-ordered product [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' CB1 (mb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' u) � OB1 8g (u) � ��� u→0 = CA0 (mb) i � ddxe−i (nx/2) umb � T � L(1) ξqsc (x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' OA0−u 8g (0) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (41) of the leading power current OA0−u 8g (0) = χu−hc(0) S† n(0) Sn(0) /n 2 /Au−hc⊥(0) (1 + γ5) S† n(0) h(0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' equal to (5) up to a replacement of the hardcollinear fields by the u-hardcollinear fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' and subleading Lagrangian L(1) ξqsc (x) = qsc(x+)S† n(0) Sn(0) � Qs /Bu−hc⊥ + /Au−hc⊥ � χu−hc(x) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (42) The jet-function J (umb) appears after integrating out the u-anti-hardcollinear quark fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We note a close resemblance to the structure of the resolved contribution, where a similar time-ordered product appears (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We find the new soft function �S (ω, ω1, ω2) which corresponds to the function S (ω, ω1, ω2) in the limit ω1 ∼ ω2 ≫ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In this limit, we can consider the light soft quarks to be softcollinear qs → qsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In this function �S (ω, ω1, ω2) higher power corrections in ω/ω1,2 are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the limit, where the momentum fractions u → 0 and u′ → 0, the jet function J (mb (p+ + ω) , u, u′) fulfills the following relation � Λ −p+ dω �J (mb (p+ + ω) , u, u′) S(ω)� = � Λ −p+ dωJg(mb(p+ + ω)) �S(ω, mbu, mbu′) , (43) where the brackets indicate that the u → 0 and u′ → 0 limits have to be taken and that the hardcollinear quark fields in J are regarded as softcollinear fields, χhc → qsc in accordance with (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' It is crucial that the soft function �S (ω, ω1, ω2) appears both in the A-type contribution in the limit ω1 ∼ ω2 ≫ ω and in the B-current term if one expands for small u and u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Before we proceed, let us comment on the structure of �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the asymptotic regime, where ω1,2 ≫ ω, we can match the �S on the leading power shape function �S (ω, ω1, ω2) = � dω′K(ω, ω′, ω1, ω2)S(ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='ghc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='shc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='Figure 3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='The SCET representations of the full theory diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1, see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The matching kernel K(ω, ω′, ω1, ω2) introduced in (44) can be computed perturbatively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' to extract K, we can replace B-meson state by a b-quark in the definition of the soft function and calculate both sides of the equation on the partonic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The LP soft function here appears since the limit ω1,2 → ∞ is equivalent to the treatment of t and t′ as infinitesimal variables in (32) and, consequently, the soft Wilson lines obtained from decoupling in the anti-hardcollinear direction Sn cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At the same time, the softcollinear quark field produces an additional soft Wilson line associated with the hardcollinear direction Sn because we require the softcollinear quark to have the same gauge transformation as a hardcollinear field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, the structure of the soft function corresponds to LP shape function S(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Consistency of the second and third refactorisation conditions, which approach the softcollinear limit from two different directions as shown in Figure 2, leads to � Λ −p+ dω �J (mb (p+ + ω) , u, u′) S(ω)� = � Λ −p+ dωJg(mb(p+ + ω)) � dω′K(ω, ω′, ω1, ω2)S(ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (45) This relation implies that the kernel K can be obtained from the quark-gluon jet function in the limit when momentum fraction of the quark tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Furthermore, it confirms that the kernel K is a perturbative object and that the softcollinear scale can be treated perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we note that softcollinear quarks must appear on both sides of the cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Fermion num- ber conservation implies that only in this case we get a non-vanishing decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Consequently, the endpoint divergences only appear in the limit when both u and u′ are small or when ω1 and ω2 are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Figure 3 shows that the A- and B-type current have the same structure in the refactorisation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' On the left, the s-quark is soft and emitted through the insertion of the subleading power Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' On the right, the s-quark is hardcollinear and emitted directly from the hard B-type vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' When the fraction of the hardcollinear momentum of the s-quark tends to zero, the B-type current refactorises into the time-ordered product represented on the left, and both diagrams rep- resent the same full theory configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This duality in the description leads to the appearance of the endpoint divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' A similar problem has already been identified in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [49,50], in the context of QED corrections in Bs → µ+µ− due to O7 operator at the amplitude level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='1 Refactorisation at leading order Based on the refactorisation conditions, we first discuss the procedure of refactorisation at the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We explicitly verify the conditions using the leading order results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Starting with the last refactorisation condition, we consider the factorisation theorem of the A-type contribution when the soft function is considered in the limit ω1 ∼ ω2 ≫ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This asymptotic limit of the soft function can be analysed by means of semi-perturbative methods [51], where the energetic softcollinear quarks are treated perturbatively, while ordinary soft modes are assumed to be nonperturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the leading order, this corresponds to the replacement of the softcollinear quarks by a cut propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We anticipate the endpoint divergence in the convolution of the soft and the anti-hardcollinear jet functions and use dimensional MS regularisation within the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We find the following expression of the asymptotic soft function at leading order [51],: �S (ω, ω1, ω2) = CFA(ϵ) αs (4π) ω1−ϵ 1 δ(ω1 − ω2) � Λ ω dω′ S(ω′) �(ω′ − ω) µ2 �−ϵ , (46) which includes the leading power shape function S(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Note that this expression, in principle, receives corrections of higher order in αs and ΛQCD/ω1,2, which we do not take into account in the leading order analysis within this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' A(ϵ) was defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (24) 9 We convolute the asymptotic soft function with the anti-hardcollinear jet functions for large ω1 and ω2 only by restricting the limits of the ω1 integral to mb and +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' These integration limits will become clear once we consider the B-type current contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Starting with the factorisation formula of the A type current given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (39), the asymptotic contribution of the A type current reads at leading order: dΓ dEγ |asy A = 2N |CA0 LO(mb)|2 � Λ −p+ dωJLO g (mb(p+ + ω)) � ∞ mb dω1JLO(ω1) � ω1 0 dω2J ∗ LO(ω2) �S(ω, ω1, ω2) = N|CA0 LO (mb) |2 αsCF (4π) mb 1 ϵA(ϵ) � Λ −p+ dω SLO(ω′) �mb(ω + p+) µ2 �−ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (47) The 1 ϵ pole is the manifestation of the endpoint divergence in the resolved contribution in the limit ω1 ∼ ω2 ≫ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the next step, we will see that the specific choice mb as a lower limit of the ω1 integration is induced by the refactorisation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The lower limit in the ω2 integral can be chosen to be non-negative due to the delta function δ(ω1 − ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Now we take the limit u → 0 in the factorisation theorem of the B-type current at leading order, which we derived in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (25) before performing the integrals over u and u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This leads to dΓ dEγ |u,u′→0 B = −N ��CA0 LO (mb) ��2 αsCF (4π) mb 1 ϵ A(ϵ) � Λ −p+ dω SLO(ω) �mb(ω + p+) µ2 �−ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (48) This result differs from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (47) only by an overall sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The sum of these two terms is finite and equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This leading-order result is a special case of the all-order relation, which follows from refactorisation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In the ω1 ∼ ω2 ≫ ω (asymptotic) limit of the A-type current (with integration limits over ω1 from mb to +∞), we exactly single out the same term as in the 9We note that we do not confirm the leading order result of the asymptotic soft function of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [4] in the dimensional regularisation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 15 u → 0 of the B-type current up to a minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This reflects the fact that in the limits u → 0 and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8 − O8 contribution have the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Moreover, we see that with the relations mbu = ω1 and mbu′ = ω2, the u, u′ → 1 limit corresponds to the limit ω1, ω2 → mb, which fixes the integration limit in the subtraction term of the A-type current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We can summarise the relation we just verified at LO as dΓ dEγ |asy A = (−1) dΓ dEγ |u,u′→0 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (49) The refactorisation conditions guarantee that the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (49) holds to all orders in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' To make this relation useful for the reshuffling of the factorisation theorem, let us consider an integral of �S(ω1, ω2, ω)J(ω1)J ∗(ω2) over the ω1,2 ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Since �S(ω1, ω2, ω) is expanded for ω1,2 ≫ ω, this integral is scaleless and equal to zero in dimensional regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We then perform the following manipulations of the integration limits 0 = � ∞ 0 dω1 � ∞ 0 dω2 = 2 � ∞ 0 dω1 � ω1 0 dω2 = 2 �� mb 0 dω1 + � ∞ mb dω1 � � ω1 0 dω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (50) In the second step, we made use of the fact that the integrand is symmetric in ω1 and ω2 as we derived to all orders in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we split the integration region into two parts suitable for the subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The term integrated over ω1 from mb to ∞ is already in the form suitable for the subtraction of the A-type term and equal to (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' To bring the second term into the form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (48), we perform substitutions ω1 = mb u and ω2 = mb u′, use (40) to replace J(ω1) by the singular part of the CB1 matching coefficient and then use the second refactorisation condition to derive 2N ��CA0 LO(mb) ��2 � mb 0 dω1JLO(ω1) � ω1 0 dω2J ∗ LO(ω2) � Λ −p+ dωJLO g (mb(p+ + ω)) �S(ω, ω1, ω2) = 2N � 1 0 du � CB1 LO (mb, u) � � 1 u du′ � CB1 LO (mb, u′) � � Λ −p+ dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� (51) We rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (49) using the functions which enter the factorisation theorem: 2N ��CA0 LO(mb) ��2 � ∞ mb dω1JLO(ω1) � ω1 0 dω2J ∗ LO(ω2) � Λ −p+ dωJLO g (mb(p+ + ω)) �S(ω, ω1, ω2) = − 2N � 1 0 du � CB1 LO (mb, u) � � 1 u du′ � CB1 LO (mb, u′) � � Λ −p+ dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (52) We see with the help of (51) that the fact that the sum of asymptotic contributions is equal to zero is a consequence of our refactorisation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' It is now clear that these two subtraction terms, which add up to zero, make it possible to reshuffle the factorisation theorem and cancel the endpoint divergences at the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='2 Bare refactorised factorisation theorem The generalisation of the LO order result to all orders is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Since we are still working in d-dimensons with bare objects, we can insert a scaleless expression into the factorisation theorem using the integral manipulations we performed at LO, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (50) Using the all-orders refactorisation conditions discussed at the beginning of this section, we then can cast the subtraction term into the following form with the help of the same manipulations as in the LO case and generalise eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (52) to all orders: 0 = 2N ��CA0 (mb) ��2 � Λ −p+ dωJg (mb (p+ + ω)) � ∞ mb dω1J (ω1) � ω1 0 dω2J ∗ (ω2) �S (ω, ω1, ω2) + 2N � 1 0 du � CB1 (mb, u′) � � 1 u du′ � CB1∗ (mb, u′) � � Λ −p+ dω �J (mb (p+ + ω) , u, u′) S(ω)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (53) Starting from the all-order bare factorisation theorem dΓ dEγ = 2N ��CA0 (mb) ��2 � ∞ −∞ dω1J (ω1) � ω1 −∞ dω2J ∗ (ω2) � Λ −p+ dωJg (mb (p+ + ω)) S (ω, ω1, ω2) + 2N � 1 0 duCB1 (mb, u) � 1 u du′CB1∗ (mb, u′) � Λ −p+ dωJ (mb (p+ + ω) , u, u′) S(ω) (54) and subtracting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (53) we arrive at dΓ dEγ |A+B = 2N � Λ −p+ dω � Jg(mb(p+ + ω)) ��CA0 (mb) ��2 (55) × � ∞ −∞ dω1 � ω1 −∞ dω2J(ω1) J ∗(ω2) � S (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �S(ω, ω1, ω2) � + � 1 0 du � 1 u du′ � CB1 LO (mb, u) CB1∗ (mb, u′) J (mb (p+ + ω) , u, u′) S (ω) − � CB1 (mb, u) � � CB1∗ (mb, u′) � �J (mb (p+ + ω) , u, u′) S(ω)� �� , where �J (mb (p+ + ω) , u, u′) S(ω)� = Jg(mb(p+ + ω)) �S(ω, mbu, mbu′) and � CB1 (mb, u′) � = (−1)CA0 (mb) mbJ (umb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We note here that the second term effectively restricts the integration range over ω1 to a finite range in the first line and consequently removes endpoint divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Thus these terms need to be added together before the ω1 integral is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Similarly, the last term removes the endpoint divergence of the third term, and therefore u integration has to be performed after these two terms are added up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' In addition, we note that the integrals in the first term are finite for large negative values of ω1 and ω2 due to nonperturbative dynamics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' At this point, the convolutions integrals in the A- and B-type contributions are no longer divergent, and we can renormalise the functions entering the factorisation theorem and take the limit d → 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='3 Refactorised factorisation theorem after renormalisation We achieved refactorisation at the level of the bare factorisation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' It has been pointed out that refactorisation and renormalisation do not commute in general [23, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Therefore, for 17 the result to be helpful for the resummation of the large logarithms, we must prove that we can express the factorisation theorem in terms of renormalised objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' To this end, we have to replace bare quantities with renormalised ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The renormalisation of hard matching coefficients is well-established CA0 bare(mb) = ZA0(µ) CA0 ren(µ, mb) , (56) CB1 bare(u) = � 1 0 du′ ZB1(µ, u, u′) CB1 ren(µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content='u′) , (57) where the one-loop renormalisation factors can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The LP jet function is renormalised according to Jbare g (p2) = � p2 o dp′2 ZJg(µ, p2 − p′2) Jren g (µ, p′2) , (58) with the ZJg factor given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [53, 54] up to the three-loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Similarly, the LP shape function Sbare(ω) = � dω′ ZS(µ, ω − ω′) Sren(µ, ω′) (59) is well-known [55] Much less is known about NLP objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The radiative jet function is a notable example which appeared before in the context of B → γℓν [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' It has recently been computed at the two-loop order in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The most important detail is that the time-like (ω > 0) and space-like (ω < 0) radiative jet functions do not mix under renormalisation J + bare/ren(ω) = θ(ω)Jbare/ren(ω) , (60) J − bare/ren(ω) = θ(−ω)Jbare/ren(ω) , (61) and J + bare(ω) = � ∞ 0 dω′ Z+ J (µ, ω, ω′) , J + ren(µ, ω′) , (62) J − bare(ω) = � 0 −∞ dω′ Z− J (µ, ω, ω′) J − ren(µ, ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (63) This separation into time-like and spec-like jet functions is necessary since we choose to integrate the subtraction term only over non-negative values of ω1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we define the renormalisation of the NLP soft and jet functions Sbare(ω, ω1, ω2) = � dω′dω′ 1dω′ 2 ZS(µ, ω, ω′, ω1, ω′ 1, ω2, ω′ 2) Sren(µ, ω′, ω′ 1, ω′ 2) , (64) Jbare(p2, u1, u2) = � dp′2 � 1 0 du′ 1 � 1 0 du′ 2 ZJ(µ, p2 − p′2, u1, u′ 1, u2, u′ 2) Jren(p′2, u′ 1, u′ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (65) These renormalisation kernels are currently unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 18 We require that A- and B-type contributions are separately RG invariant (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' [56] for analogous treatment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This leads to the following conditions on the renormalisation kernels |ZA0|2 � dω � dω1 � dω2 ZJg(ω − ω′)ZJ(ω1, ω′ 1) Z† J(ω2, ω′ 2) ZS(ω, ω′′, ω1, ω′′ 1, ω2, ω′′ 2) =δ(ω′ − ω′′) δ(ω′ 1 − ω′′ 1) δ(ω′ 2 − ω′′ 2) , (66) and � 1 0 du1 � 1 0 du2 � dω ZB1(u1, u′ 1) ZB1†(u2, u′ 2) ZJ(ω − ω′, u1, u′ 1, u2, u′ 2) ZS(ω − ω′′) = δ(ω′ − ω′′) δ(u′ 1 − u′′ 1) δ(u′ 2 − u′′ 2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (67) and further, RG invariance of the subtraction term leads to |ZA0|2 � ∞ 0 dω1 � ∞ 0 dω2 � dω Z+ J (ω1, ω′ 1) Z+† J (ω2, ω′ 2) ZJg(ω − ω′) Z�S(ω − ω′′, ω1, ω′′ 1, ω2, ω′′ 2) = δ(ω′ − ω′′) δ(ω′ 1 − ω′′ 1) δ(ω′ 2 − ω′′ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (68) These conditions are sufficient to prove that renormalisation and refactorisation commute and there is no leftover term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We can now insert the above definitions into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' (55), dΓ dEγ |A+B = 2N � Λ −p+ dω � Jren g (mb(p+ + ω)) ��CA0 ren (mb) ��2 (69) × � ∞ −∞ dω1 � ω1 −∞ dω2J + ren(ω1) J +∗ ren(ω2) � Sren (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �Sren(ω, ω1, ω2) � + � 1 0 du � 1 u du′ � CB1 ren (mb, u) CB1∗ ren (mb, u′) Jren (mb (p+ + ω) , u, u′) Sren (ω) − � CB1 ren (mb, u) � � CB1∗ ren (mb, u′) � �Jren (mb (p+ + ω) , u, u′) Sren(ω)� �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This is our final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Endpoint divergences are manifestly absent, assuming one performs the integrals over ω1 after adding the first and second terms together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Similarly, the integrals over u should be performed after adding the last two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This renormalised factorisation theorem allows for a consistent resummation of large logarithms within the resolved O8 − O8, using standard RG methods owing to the fact that each object appearing in the above equation is a single scale object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' However, a judicious choice of scale might be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' 19 5 Summary and Outlook In the present paper, we identified the divergences in the resolved, but also in the direct subleading O8 − O8 as endpoint divergences which lead to a breakdown of the factorisation theorem already at leading order in four space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The failure of naive factorisation does not allow for consistent separation of scales and, consequently, resummation of large logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' However, it was recently shown [9] that the resolved contributions still represent the most significant uncertainty in the inclusive ¯B → Xsγ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Large scale dependence and also a large charm mass dependence were identified in the lowest order result of the resolved contribution, which calls for a systematic calculation of αs corrections and RG summation of all resolved contributions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' A mandatory input for this task is a well-defined factorisation formula for these subleading corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This critical step was established in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' The next step consists of computing renormalisation kernels for the NLP soft and jet functions, extracting the anomalous dimensions and solving the RG equations to resum large logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Recent intensive studies of the power corrections in collider applications of SCET [19,22–24, 42,52] lead to the development of new techniques that allow for a reshuffling of terms within the factorisation formula so that all endpoint divergences cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' We used these new techniques in our flavour application which includes nonperturbative functions typically not present in collider applications of SCET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Unlike in the h → γγ decay [22], in the considered SCETI problem, there are no leftover terms present after renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' To derive a consistent factorisation theorem, we first established the bare factorisation theorem for the resolved and direct contributions on the operatorial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Then we derived the all-orders refactorisation conditions applicable to our process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' This idea is based on the fact that in certain limits, the two terms of the subleading O8 − O8 contribution have the same structure, which guarantees that the endpoint divergences cancel between the two terms to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Finally, we proved that we could express the factorisation theorem in terms of renormalised objects so that the result can be used for the resummation of the large logarithms within the resolved contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' Acknowledgements We thank Martin Beneke and Matthias Neubert for their valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' RS would also like to thank Mathias Garny and Jian Wang for many discussions on power corrections in SCET and Mikolaj Misiak for a discussion on the theoretical predictions for B → Xsγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' TH is grateful to Michael Benzke for uncounted discussions on the resolved contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' RS is supported by the United States Department of Energy under Grant Contract DESC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' TH is supported by the Cluster of Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter" (PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 39083149), as well as by the BMBF Verbundprojekt 05H2018 - Belle II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfxf6F/content/2301.01739v1.pdf'} +page_content=' TH also thanks the CERN theory group for its hospitality during his regular visits to CERN where part of the work was done.' metadata={'source': 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a/INE1T4oBgHgl3EQfXwS1/content/tmp_files/2301.03131v1.pdf.txt b/INE1T4oBgHgl3EQfXwS1/content/tmp_files/2301.03131v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c593ecf8c822b50150af631576b34223a4bce18 --- /dev/null +++ b/INE1T4oBgHgl3EQfXwS1/content/tmp_files/2301.03131v1.pdf.txt @@ -0,0 +1,1508 @@ +arXiv:2301.03131v1 [math.AT] 9 Jan 2023 +INTRINSIC CONVERGENCE OF THE HOMOLOGICAL TAYLOR TOWER FOR +r-IMMERSIONS IN Rn +GREGORY ARONE AND FRANJO ˇSARˇCEVI´C +Abstract. For an integer r ≥ 2, the space of r-immersions of M in Rn is defined to be the +space of immersions of M in Rn such that at most r − 1 points of M are mapped to the same +point in Rn. The space of r-immersions lies “between” the embeddings and the immersions. +We calculate the connectivity of the layers in the homological Taylor tower for the space of r- +immersions in Rn (modulo immersions), and give conditions that guarantee that the connectivity +of the maps in the tower approaches infinity as one goes up the tower. We also compare the +homological tower with the homotopical tower, and show that up to degree 2r − 1 there is a +“Hurewicz isomorphism” between the first non-trivial homotopy groups of the layers of the two +towers. +Contents +1. +Introduction +1 +2. +Prerequisites +5 +2.1. +Cubical diagrams +5 +2.2. +Manifold calculus of functors +6 +2.3. +Spectra +9 +3. +The homological Taylor tower for reduced r-immersions in Rn +12 +4. +r-configuration spaces in Rn as complements of subspace arrangements +13 +5. +Total fiber of a retractive cubical diagram +16 +6. +The cube of r-configuration spaces is retractive +18 +7. +Connectivity of the cube of (co)homologies of r-configuration spaces +19 +8. +Convergence result +22 +9. +Comparing with the unstable tower +23 +10. +Further questions +26 +References +27 +1. Introduction +Let M be a smooth manifold of dimension m, and fix an integer r ≥ 2. An r-immersion of M +in Rn is an immersion of M in Rn such that the preimage of every point in Rn contains at most +r − 1 points of M. The space of r-immersions of M in Rn is denoted by rImm(M, Rn). For +2020 Mathematics Subject Classification. Primary: 57R42; Secondary: 55R80, 57R40, 55P42. +Key words and phrases. calculus of functors, manifold calculus, Taylor tower, embeddings, immersions, r- +immersions, homotopy of spectra, homological convergence, partial configuration space. +Acknowledgements. F. ˇSarˇcevi´c was partially supported by the grant P20 01109 (JUNTA/FEDER, UE). +1 + +r = 2, 2-immersions are the same thing as injective immersions, which are essentially the same +as embeddings in nice cases. In any case, we have inclusions of subspaces +Emb(M, Rn) ⊆ 2 Imm(M, Rn) ⊂ 3 Imm(M, Rn) ⊂ · · · ⊂ rImm(M, Rn) ⊂ · · · ⊂ Imm(M, Rn). +In this paper we study the homological Taylor tower of the r-immersions functor. The “Taylor +tower” is meant in the sense of manifold calculus (also known as embedding calculus) developed +by Weiss [Wei99] and Goodwillie-Weiss [GW99]. +The basic idea of manifold calculus is the following. In order to study the homotopy type of a space +such as rImm(M, Rn), one views it as a particular value of the presheaf rImm(−, Rn) defined on +M (one can also consider more general target manifolds than Rn, but we will content ourselves +with maps into Rn). A presheaf is a contravariant functor on the poset O(M) of open subsets +of M. Inside O(M) there is a sequence of subposets O1(M) ⊂ · · ·Ok(M) ⊂ · · · ⊂ O∞(M), +where Ok(M) is the poset of open subsets of M that are diffeomorphic to the disjoint union of +at most k copies of Rm. By restricting a presheaf F to Ok(M) and then extrapolating back to +O(M) one obtains a tower of approximations to F, which is usually denoted as follows +F → (T∞F → · · · → TkF → Tk−1F → · · · T0F). +This is called the “Taylor tower” of F. Manifold calculus, and the Taylor tower in particular, has +had many consequences and applications [Mun05], [Vol06], [ALV07], [Mun11], [DH12], [ST16], +[BdBW18]. +In this paper we investigate the Taylor tower that calculates the homology of the space rImm(M, Rn). +In practice, this means the following. First of all, it is convenient to replace the space of r- +immersions with r-immersions modulo immersions. Let us suppose that we fix a basepoint in +Imm(M, Rn), and let rImm(M, Rn) be the homotopy fiber of the inclusion map rImm(M, Rn) → +Imm(M, Rn). Let HZ denote the Eilenberg-MacLane spectrum. We are interested in the Taylor +tower of the presheaf of Spectra, defined by the formula +U �→ HZ ∧ rImm(U, Rn). +(more precise definitions are given in Section 2). +Our main result concerns the rate of convergence of the Taylor tower of this functor. +The +question of convergence is a fundamental one. +We will distinguish between two aspects of +convergence: how strongly the tower converges to its limit, and what it converges to. We will +say that the Taylor tower of a functor F converges intrinsically at M if the connectivity of the +map TkF(M) → Tk−1F(M) approaches ∞ as k approaches ∞. We say that the Taylor tower +of F converges strongly to F(M) if the connectivity of the map F(M) → TkF(M) approaches +∞ as k approaches ∞. Strong convergence implies intrinsic convergence, but the converse does +not have to be true. In practice it seems that for “natural” functors that we know, whenever the +Taylor tower of F converges intrinsically, it converges strongly to F. But intrinsic convergence +is usually much easier to prove than strong convergence. +Before we state our main result, let us recall, for context, that one of the deepest results in +functor calculus is the Goodwillie-Klein-Weiss convergence theorem [GW99], [GK08], [GK15]. +Theorem 1.1 (Convergence of the Taylor tower for spaces of embeddings). If M is a smooth +closed manifold of dimension m, and N is a smooth manifold of dimension n, then the map +Emb(M, N) → Tk Emb(M, N) +2 + +is +k(n − m − 2) + 1 − m-connected. +In particular, if n−m−2 > 0, then the connectivities grow with k and the Taylor tower therefore +converges strongly to Emb(M, N). +There is an easier, but also important convergence result for the homological version of the tower, +which is more directly relevant to this paper. Define Emb(M, Rn) to be the homotopy fiber of +the inclusion Emb(M, Rn) → Imm(M, Rn). Consider the contravariant functor from O(M) to +Spectra that sends U to HZ ∧ Emb(U, Rn). This functor represents the homology of the space +of embeddings modulo immersions. The Taylor tower of this functor is known to converge when +n > 2m + 1 [Wei04]. +Now let us state our main result +Theorem 1.2. Let M be m-dimensional. Assume that n ≥ 2. If r ≤ n + 1, the Taylor tower +for HZ ∧ rImm(M, Rn) converges intrinsically when +n > rm + 1 +r − 1 . +If r ≥ n + 1 then the Taylor tower converges intrinsically when n > m + 1. +Remarks 1.3. +(1) When r = n + 1 the two statements are equivalent. +Indeed, the function f(n) = +n2 − nm − m − 1, n ∈ N, is positive only for n > m + 1. +(2) When r = 2 we get the condition n > 2m + 1, which is the known condition for the +convergence of the Taylor tower of HZ ∧ Emb(M, Rn). +(3) The condition n > rm+1 +r−1 is equivalent to rm−(r−1)n < −1. The number rm−(r−1)n +equals, at least when it is positive, to the dimension of the intersection of r copies of Rm +embedded in Rn in a general position. +Next let us discuss the proof. Let F be a presheaf defined on a suitable category of m-dimensional +manifolds and codimension zero embeddings. The basic building blocks in the construction of +the Taylor tower of F are spaces of the form F(� +i Rm), for i = 0, 1, 2, . . .. The homotopy fiber +of the map TkF → Tk−1F depends on the total homotopy fiber of the following cubical diagram, +indexed by the poset of subsets of k = {1, . . . , k}: +(1) +S �→ F + +� +k\S +Rm + + +This homotopy fiber is sometimes called the k-th derivative (or the k-th cross-effect) of F at +∅. The following fact is particularly important for analysing intrinsic convergence. Recall that a +cubical diagram is called c-cartesian if the map from the initial object to the homotopy limit of the +rest of the cubical diagram is c-connected. Suppose the cubical diagram (1) is ck-cartesian. Then +the map TkF(M) → Tk−1F(M) is ck − mk-connected. Thus the Taylor tower of F converges +intrinsically at M if the number ck − mk approaches ∞ as k approaches ∞. +When F(M) = Emb(M, Rn), there is a well-known equivalence Emb(� +k Rm, Rn) ≃ Conf(k, Rn), +where Conf(k, Rn) is the configuration space of ordered k-tuples of pairwise distinct points in Rn. +3 + +Similarly, there is an equivalence between rImm(� +k Rm, Rn) and the so-called r-configuration +space, also called no r-equal configuration space, defined by +rConf(k, Rn) := rImm(k, Rn). +This is the space of ordered k-tuples of points in Rn where at most r −1 are allowed to be equal. +A proof of the equivalence +rImm( +� +k +Rm, Rn) +≃−→ rConf(k, Rn) +is given in [AˇS22]. Thus r-configuration spaces are basic building blocks in the Taylor tower of +rImm(M, Rn). +To analyse the intrinsic convergence of the Taylor tower of the functor HZ ∧ rImm(−, Rn), one +needs to calculate how cartesian the following k-dimensional cubical diagram is +(2) +S �→ HZ ∧ rConf(k \ S, Rn). +The space rConf(i, Rn) is the complement of a subspace arrangement in Rni. It follows that the +homology of r-configuration spaces is accessible by means of the Goresky-MacPherson formula +and other such tools. The homology of r-configuration spaces was studied by a number of people, +starting with Bj¨orner and Welker [BW95]. +Using the Goresky-MacPherson formula and the results in [BW95] we prove the following result +(it is combining Proposition 7.7 and Theorem 8.1) +Theorem 1.4. When r ≤ n + 1, the cube (2) is k(n − 1) + +� k +r +� +(r − n − 1)-cartesian, and the +map +pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn) +is +k +� +nr − 1 +r +− m − 1 +r +� +− (k mod r) +r +(r − n − 1)-connected. +Here (k mod r) := k − r +� k +r +� +. +When r ≥ n + 1, the cube (2) is k(n − 1) + r − n − 1-cartesian, and the map pk is +k(n − m − 1) + r − n − 1-connected. +Theorem 1.2 follows easily from Theorem 1.4. +In Section 9 we compare the tower of the homological functor HZ ∧ rImm(M, Rn) with that of +the tower of the homotopical functor rImm(M, Rn). Let us suppose that we chose a basepoint in +the space rImm(M, Rn). In this case the presheaf rImm(−, Rn) takes values in pointed spaces, +and we have the following diagram of presheaves: +(3) +rImm(−, Rn) +i←− rImm(−, Rn) +h−→ Ω∞HZ ∧ rImm(−, Rn). +It is well-known that the map i induces an equivalence of all layers except the first one. Indeed, +the map i is the homotopy fiber of the map from rImm(−, Rn) to its linear approximation. Thus +we can view the map h as a map from the higher layers/derivatives of rImm(−, Rn) to the +corresponding layers/derivatives of Ω∞HZ∧rImm(−, Rn), which are essentially the same as the +layers/derivatives of HZ ∧ rImm(−, Rn), since Ω∞ commutes with Taylor approximations. +4 + +When r = 2, the second derivative of rImm(−, Rn) is equivalent to Sn−1, and the second +derivative of HZ ∧ rImm(−, Rn) is HZ ∧ Sn−1. It follows that in the case r = 2, the map h +in (3) induces the Hurewicz homomorphism from the second derivatives of rImm(−, Rn) to the +second derivative of HZ ∧ rImm(−, Rn). In particular, it follows that the connectivity of the +quadratic layers of the Taylor towers of rImm(−, Rn) and of HZ ∧ rImm(−, Rn) is the same, +and their first non-trivial homotopy groups are isomorphic. +By contrast, at degrees higher than 2, the layers of the homotopical tower rImm(−, Rn) and of +the homological tower of the functor HZ ∧ rImm(−, Rn) have different connectivities, and there +is no Hurewicz type isomorphism between them. +And again by contrast, in Section 9 we show that for r > 2 the map h in diagram (3) induces +a Hurewicz type isomorphism between first non-trivial homotopy groups of layers roughly up to +degree 2r − 1. See Theorem 9.1 for precise statement. +Organization of the paper. In Section 2 we review some background material on cubical diagrams, +manifold calculus and spectra. In Section 3 we introduce the homological Taylor tower that is +the main subject of this paper. +In Section 4 we make an excursion into the subspace arrangements. We describe r-configuration +spaces via subspace arrangements and compute their cohomology using the Goresky-MacPherson +theorem. +In Section 5 we define the notion of a retractive cubical diagram. This is a diagram where the +maps have sections that satisfy a certain hypothesis. We prove that the homotopy groups of +the total homotopy fiber of a retractive cube are isomorphic to the total kernel of the cube of +homotopy groups. +In Section 6 we prove that the cube of r-configuration spaces that controls the layers in the Taylor +tower is retractive. In Section 7 we prove our main result about the homological connectivity +of the cube of r-configuration spaces. In Section 8 we prove the main result about the intrinsic +convergence of the Taylor tower of HZ ∧ rImm(M, Rn). +In Section 9 we compare the tower of HZ ∧ rImm(M, Rn) with the tower of rImm(M, Rn) in +low degrees. We prove that the layers in the two towers have the same connectivity up to degree +2r − 1 (with some exceptions in the cases r = 2, 3). +In Section 10 we discuss some possible directions for further exploration. +2. Prerequisites +2.1. Cubical diagrams. Cubical diagrams play an important role in functor calculus, and in this +paper in particular, so we will recall a few elementary facts about them. All the results in this +subsection, and much more, can be found in [Goo92]. +Let k denote the standard set with k elements {1, . . . , k}. Let P(k), or just P(k), denote the +poset of subsets of k. A k-dimensional cubical diagram in a category C is a functor χ: P → C. +It is easy to see that P(k) is equivalent to P(k)op, so a contravariant functor from P(k) to C is +called a cubical diagram as well. We will mostly consider cubical diagrams in (pointed) spaces +and spectra, and also in abelian groups. +5 + +Given a cubical diagram χ in topological spaces or spectra, there is a natural map +iχ : χ(∅) → holim +∅̸=S⊂k χ(S). +We say that χ is c-cartesian, if this map is c-connected. The homotopy fiber of this map is called +the total homotopy fiber of χ. The total homotopy fiber of χ is denoted by tfiber(χ). Clearly if +χ is c-cartesian then tfiber(χ) is c−1-connected. The converse always holds for cubical diagrams +of spectra, and it holds for spaces under the additional assumption that iχ is surjective on path +components. +One can identify a k-dimensional cubical diagram with a map of two k − 1-dimensional cubical +diagrams. Given a k-dimensional cubical diagram χ, let us define two k − 1-dimensional cubical +diagrams χ1 and χ2 as follows: χ1(U) = χ(U), and χ2(U) = χ(U ∪ {k}). Then χ can be +identified with the map of cubes χ1 → χ2. Furthermore, there is a homotopy fibration sequence +whose meaning is that total homotopy fiber can be calculated as an iterated homotopy fiber +tfiber(χ) ≃ hofiber(tfiber(χ1) → tfiber(χ2)). +When χ is a cubical diagram of abelian groups, we define the total kernel of χ to be +tkernel(χ) := ker(χ(∅) → +k +� +i=1 +χ({i})). +Just as with total fibers, the total kernel can be calculated as an iterated kernel. There is a +natural isomorphism +tkernel(χ) ∼= ker(tkernel(χ1) → tkernel(χ2)). +When χ is a cubical diagram of spaces or spectra, there is a natural homomorphism of graded +groups +π∗(tfiber χ) → tkernel(π∗χ). +This homomorphism is not an isomorphism in general. In Section 5 we will investigate a condition +on a cubical diagram that guarantees for it to be an isomorphism. +2.2. Manifold calculus of functors. Let M be a smooth manifold of dimension m. Define +O(M) to be the poset category of open subsets of M. Objects of O(M) are open sets U ⊆ M, +and morphisms U → V are the inclusions U ⊆ V . +Manifold calculus of functors, developed by Weiss [Wei99] and Goodwillie-Weiss [GW99], studies +contravariant functors from O(M) to a category that supports a reasonable notion of homo- +topy. In their foundational papers, Goodwillie and Weiss only considered functors with values in +topological spaces, and maybe spectra. Nowadays it is natural to let the target category to be +an ∞-category. We will content ourselves with functors with values in (pointed) spaces and in +spectra. +Technically speaking, manifold calculus applies to functors that are good, in the sense that they +satisfy the following two conditions: +(i) they are isotopy functors, and +(ii) they are finitary. +A functor is an isotopy functor if it takes isotopy equivalences to weak homotopy equivalences +(for the definition of isotopy equivalence see [MV15, Definition 10.2.2]). It is finitary if for every +6 + +monotone union � +i Ui (where Ui ⊂ Ui+1 for i = 1, 2, ...) the canonical map from F(� +i Ui) to +holimi F(Ui) is a weak homotopy equivalence. +If F is a ”half-good” contravariant functor (cofunctor), i.e. an isotopy functor which is not a +finitary functor, then we need to tame this functor. We call V ∈ O(M) tame if V is the interior +of a compact smooth codimension zero submanifold of M. As mentioned in [GKW01], property +(ii) ensures that a good cofunctor F on O(M) is essentially determined by its behavior on tame +open subsets of M. +In particular, suppose F is a cofunctor from O(M) to Top having property (i). Then the functor +defined by +F #(V ) := holimtame U⊂V F(U) +for V ∈ O(M) has also property (ii), i.e. F # is a good cofunctor on O(M). We call F # the +taming of F. +There exists a natural transformation F → F #. The map F(V ) → F #(V ) is an equivalence +whenever either F or V is tame. +The motivating example for the development of the manifold calculus of functors is the embedding +functor. +Definition 2.1. (Space of embeddings) Let M and N be smooth manifolds. +• A smooth embedding of M in N is a smooth map f : M → N such that +1. the map of tangent spaces +Dxf : TxM → Tf(x)N +is an injection for all x ∈ M, i.e. the derivative of f is a fiberwise injection, and +2. f : M → f(M) is a homeomorphism. +• The space of embeddings, Emb(M, N), is the subspace of the space of smooth maps +from M to N consisting of smooth embeddings of M in N. The space Emb(M, N) is +topologized using Whitney C∞-topology; for an explanation see [MV15, Appendix A.2.2]). +An important example of a space of embeddings with very rich theory is the space of classical +knots defined to be the space Emb(S1, R3). +Definition 2.2. (Embedding functor) +For a smooth n-dimensional manifold N, the embedding functor Emb(−, N) : O(M) → Top is +a contravariant functor given by U �→ Emb(U, N). +The contravariance follows from the fact that an inclusion of open subsets of a manifold M gives +a restriction map of embedding spaces of manifolds. +A related notion is the space of immersions Imm(M, N), which is a space of smooth maps +f : M → N such that just the derivative of f is a fiberwise injection, (property 1. +from +Definition 2.1). If M is a compact manifold and f is an injective immersion M → N, then f is +an embedding. +The corresponding functor is the immersion functor Imm(−, N) : O(M) → Top given by +U �→ Imm(U, N). +Functors Emb(−, N) and Imm(−, N) are examples of good functors (see [Wei99] and [GKW01]). +7 + +The idea of the manifold calculus of functors is to approximate a good functor with simpler, +polynomial functors. +Definition 2.3. (Polynomial functor) +A good contravariant functor F : O(M) → Top is called polynomial of degree ≤ k if for all +U ∈ O(M) and for all pairwise disjoint closed subsets A0, ..., Ak ⊂ U, the (k + 1)-cube +P(k + 1) → Top +S �→ F(U − +� +i∈S +Ai) +is homotopy cartesian; equivalently, the map F(U) → holimS̸=∅ F(U − � +i∈S Ai) is a homotopy +equivalence. Here P(k + 1) is the poset category of all subsets of the set k + 1 = {1, ..., k + 1} +with ⊂ as the relation of partial order. Its shape is an (k + 1)-dimensional cubical diagram. +It is well known that a polynomial f : R → R of degree k such that f(0) = 0 is uniquely +determined by its values on k distinct points. In analogy, a polynomial functor is completely +determined by its values on the category of at most k open discs. +[Mun10] provides more +analogies between the ordinary calculus of functions and the manifold calculus of functors. +More precisely, let Ok(M) be the full subcategory of M consisting of open subsets of M diffeo- +morphic to ≤ k disjoint discs. We have the following theorem due to Weiss ([Wei99, Theorem +5.1]). +Theorem 2.4. Suppose F, G : O(M) −→ Top are good functors that are polynomials of degree +≤ k. If T : F → G is a natural transformation that is an equivalence for all U ∈ Ok(M), then +T is an equivalence for all U ∈ O(M). +Example 2.5. +• The functor U �→ Imm(U, N) is a polynomial of degree ≤ 1. +• The functor U �→ Emb(U, N) is not a polynomial of degree ≤ k for any k. +For the details, see [MV15, Example 10.2.10], [Wei99, Example 2.3], [Mun10, Examples 4.7 and +4.8]. +Definition 2.6. (Polynomial approximations) +For a good functor F, define for each U ∈ O(M) the kth polynomial approximation of F to be +TkF(U) = holimV ∈Ok(U) F(V ). +As Weiss proved in [Wei99, Theorems 3.9. and 6.1], such defined TkF is polynomial of degree +≤ k. Also, higher derivatives of such defined polynomial functors vanish and derivatives of a +functor and derivatives of its kth polynomial approximation agree up to kth degree, where the +derivatives of functors are defined as follows: +Definition 2.7. (Derivative of a functor) +Let Dm +1 , ..., Dm +k be pairwise disjoint open discs in M. Define a k-cube of spaces by the rule +S �→ F(� +i/∈S Dm +i ). We define the kth derivative of F at the empty set, denoted F (k)(∅), to be +the total homotopy fiber of the cube S �→ F(� +i/∈S Dm +i ). +8 + +For example, the 1st derivative of embeddings are immersions. Also, the linearization of the space +of embeddings is the space of immersions, namely there exists an equivalence T1 Emb(−, N) ≃ +Imm(−, N) ([Wei99]). +For more details and intuition behind this, see Munson’s survey [Mun10]. For other relevant +results, see [MV15, Theorem 10.2.16] and [Wei99]. +The inclusion Ok−1(U) → Ok(U) induces a map TkF(U) → Tk−1F(U) and so we obtain a tower +of functors, called the manifold calculus Taylor tower of F: +(4) +F(−) +�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ +� +�▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +�❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +T0F(−) +· · · +� +Tk−1F(−) +� +TkF(−) +� +· · · +� +T∞F(−) +� +Here T∞F denotes the homotopy inverse limit of this tower. TkF is also called the kth stage of +the Tower. +By evaluating diagram (4) on U ∈ O(M), we get a diagram of spaces with maps between the +stages that are fibrations. In particular, we can set U = M. +Definition 2.8. (Layer) +Define the kth layer of the manifold calculus Taylor tower of F to be the homotopy fiber of the +map between two successive stages of the tower, that is, +LkF = hofiber(TkF → Tk−1F). +We need to work here with a based Taylor tower. It can be accomplished by choosing a basepoint +in the space F(M) which then also bases the spaces TkF(U) for all k and U. +One of the fundamental results, which is a consequence of the Classification of homogeneous +functors theorem ([Wei99, Theorem 8.5], see also [MV15, Theorem 10.2.23 and Proposition +10.2.26]) is the following +Proposition 2.9. For a good functor F defined on m-dimensional manifolds, if the cube +S �→ F + +� +k\S +Dm + + +is ck-cartesian, then the map TkF(M) → Tk−1F(M) is ck − km-connected. More generally, if +U has handle dimension j, then the map TkF(U) → Tk−1F(U) is (ck − kj)-connected. +For the definition of handle dimension, see [MV15, Appendix A.2.1]. +It follows that the Taylor tower of F converges intrinsically at M if the number ck−mk approaches +∞ as k approaches ∞. +2.3. Spectra. The subject of this paper is a functor that represents homology. We had a choice +between working with chain complexes and the singular chains functor, or working with spectra +and using smash product with the Eilenberg-MacLane spectrum to represent homology. We chose +the latter. +9 + +We adopt a naive, old-fashioned view of spectra as sequences of spaces equipped with structure +maps between them. +Definition 2.10. (Spectrum) +A spectrum E is a sequence of based spaces {En}n∈N0 together with basepoint-preserving maps +(called structure maps) +(5) +ΣEn → En+1, +or, equivalently, the maps +(6) +En → ΩEn+1, +where Σ and Ω denote suspension and loop space, respectively. +If the maps (6) are weak equivalences, then E is called an Ω-spectrum. +Each En from an +Ω-spectrum is called an infinite loop space. +Example 2.11. (Eilenberg-MacLane spectrum) +Let n be an arbitrary positive integer and G be an arbitrary group, abelian for n > 1. Then there +exists a CW complex X such that +(7) +πn(X) ∼= G and πk(X) is trivial for k ̸= n. +A topological space X with property (7) is called an Eilenberg-MacLane space K(G, n). For +example, K(Z, 1) ≃ S1. +For an abelian group G, the Eilenberg-MacLane spectrum, denoted by HG, is defined to be the +spectrum {En}n∈N0 with En = K(G, n + 1) and maps +(8) +K(G, n + 1) → ΩK(G, n + 2). +The maps (8) are weak equivalences, hence HG is an Ω-spectrum. +Since for a spectrum E there exist maps +πi+n(En) → πi+n+1(En+1) +(for details, see [Hat02, Section 4.F]), it makes sense to define the ith homotopy group of the +spectrum E as +πi(E) = colimn πi+n(En). +Definition 2.12. A map of spectra f : E → F is a collection of maps +fn : En → Fn, n ≥ 0 +that commute with the structure maps in E = {En} and F = {Fn}. +Taking spectra as objects and maps of spectra as morphisms we can define the category of +spectra. It is denoted by Spectra. +A spectrum can be smashed with a pointed space. +Definition 2.13. Let E = {En} be a spectrum and X be a based space. The spectrum E ∧ X +is defined by +(E ∧ X)n = En ∧ X. +10 + +Since Σ(En ∧ X) ∼= (ΣEn) ∧ X, the structure maps in the spectrum E ∧ X are the products of +structure maps in E and the identity map. For a spectrum E∧X the homotopy groups πi(E∧X) +are the groups colimn πi+n(En∧X). These groups define a generalized reduced homology theory, +determined by E. +The following result is a consequence of Proposition 4F.2 in [Hat02]. See also [Whi62] for more +details on representing generalized homology theories with spectra. +Proposition 2.14. For the Eilenberg-MacLane spectrum HZ there exists an isomorphism +πi(X ∧ HZ) ∼= �Hi(X; Z). +If a spectrum E = {En}n≥0 is an Ω-spectrum, then πn(E) is +πn(E) = +� +πn(E0), +for n ≥ 0 +π0(E−n), +for n ≤ 0 +Let us note that smash product with a spectrum can be extended from pointed to unpointed +spaces. +Definition 2.15. Let E be a spectrum and X an unpointed space. Define the smash product +of E and X to be the homotopy fiber of the map +E ∧ X+ → E +induced by the canonical map X+ → S0. +For any choice of basepoint in X, there is a canonical equivalence between the new and the old +definition E ∧ X. But the new definition does not depend on a choice of basepoint. This is a +variant of the fact that reduced homology can be defined as relative homology to a basepoint, +but also can be defined independently of basepoint, using the augmented chain complex. +However, it is also important to note that without a choice of basepoint in X, there is no natural +map X → Ω∞HZ ∧ X representing the Hurewicz homomorphism. Such a map is defined only +with a choice of basepoint. +We can assume that each spectrum is an Ω-spectrum up to weak equivalence. Precisely, the +following result holds. +Proposition 2.16. Every spectrum is weakly equivalent to an Ω-spectrum. +If two spectra E and F are weak equivalent, we write E ≃ F. +Operation Σ∞ which assigns to a based space X its suspension spectrum Σ∞X, defined by +En = ΣnX with identities as structure maps, is a functor +Σ∞ : Top∗ → Spectra. +Its adjoint functor +Ω∞ : Spectra → Top∗ +is defined to be the functor which takes a spectrum E = {En}n≥0, then replaces it by an +equivalent Ω-spectrum F = {Fn}n≥0 (which exists using proposition 2.16) and finally picks off +the first place F0. In short, Ω∞(E) = F0 where F = {Fn}n≥0 ≃ E. This F0 is an infinite loop +space, which explains the notation. +11 + +It follows from the results and comments above that nth homotopy group of a spectrum E equals +the nth homotopy group of the space Ω∞(E). +Finally, let us mention that in addition to the smash product of a spectrum with a space, there is +a very important notion of smash product of spectra. For our purposes, the most naive version +of the construction suffices. Given two spectra E = {En} and F = {Fn}, we define their smash +product E ∧ F by the formulas (E ∧ F)2n = En ∧ Fn, and (E ∧ F)2n+1 = En+1 ∧ Fn, with +the structure maps being induced from the structure maps in E and F in the obvious way. The +sphere spectrum is the unit (up to homotopy) for this smash product. +One feature of smash product of spectra that plays a role in this paper is that unlike smash +product of spaces, smash product with a fixed spectrum commutes with finite homotopy limits +of spectra. More generally, it commutes with homotopy limits over a category whose classifying +space is compact. This is discussed in some detail in [LRV03]. The significance for us is that +if F is a good presheaf of spectra on M, and E is a fixed spectrum, then there are natural +equivalences +E ∧ TkF ≃ TkE ∧ F +and +E ∧ LkF ≃ LkE ∧ F. +3. The homological Taylor tower for reduced r-immersions in Rn +The main goal of this paper is to give a convergence result about the homological Taylor tower +for the space of r-immersions of a smooth manifold M in Rn. As is often the case, when studying +the homological tower, it is convenient to replace the functor of r-immersions by r-immersions +“modulo immersions”. This enables us to express the layers in the Taylor tower in terms of +r-configuration spaces. +Let M be a smooth manifold. Assume that a basepoint in the space Imm(M, Rn) is chosen, and +therefore the functor U �→ Imm(U, Rn) is a presheaf of pointed spaces on M. Recall that for +U ⊂ M, rImm(U, Rn) denotes the homotopy fiber of the map rImm(U, Rn) → Imm(U, Rn). +Let HZ denote the Eilenberg-Mac Lane spectrum. The functor +X �→ HZ ∧ X +represents reduced homology, in the sense that there is a natural isomorphism +(9) +π∗(HZ ∧ X) ∼= �H∗(X; Z). +Furthermore, recall that the functor can be extended to unpointed spaces, by defining HZ ∧ X +for unpointed X to be the homotopy fiber of the map HZ ∧ X+ → HZ. In this paper we study +the following functor +HZ ∧ rImm(−, Rn): +O(M) +→ +Spectra +U +�→ +HZ ∧ rImm(U, Rn) +This functor is representing the homology of rImm(−, Rn). +12 + +Remark 3.1. Instead of using spectra and the functor HZ ∧ − to represent homology, we could +have used chain complexes and the singular chains functor. One reason for choosing spectra is +their topological nature. The category of spectra, and of HZ-module spectra, is tensored and +cotensored over topological spaces, while the category of chain complexes is not. Of course, +this is a minor technical issue that can be overcome, but anyway it was one reason for us to +work with HZ-modules rather than chain complexes. Another reason is that working with HZ- +modules readily points to generalizations. In particular, most of our results about the functor +HZ ∧ rImm(−, Rn) can be extended to the functor Σ∞rImm(−, Rn), which in turn can be used +to obtain information about the unstable Taylor tower of rImm(−, Rn). +Remark 3.2. In [GKW01] and [Wei04], Goodwillie, Weiss and Klein point out that for a con- +travariant functor F : O(M) → Top, the cofunctor λJF given by +U �→ F(U)+ ∧ J +for a fixed spectrum J is only ”half-good”, even if F is good. Namely, it is an isotopy functor +but it fails to be finitary. As mentioned in Section 2.2, to fix this they suggest to use the taming +of λJF. We will denote the taming of a functor such as λJF by λJF #. The functor λJF # is +a good cofunctor, and there is a natural transformation λJF → λJF #, which is an equivalence +when evaluated on a tame subset of M, where by a tame subset we mean an open subset which +is diffeomorphic to the interior of a compact manifold with boundary. From now on, whenever +we write HZ ∧ rImm(−, Rn) we really mean the taming of this functor. In practice it makes no +difference since we only are interested in evaluating our functors on tame manifolds. +So we need to figure out the connectivity of the kth layer of the Taylor tower for the space +HZ∧rImm(M, Rn). By Proposition 2.9, this is determined by the homotopy fiber of the cubical +diagram, indexed by subsets of {1, . . . , k}, +S �→ HZ ∧ rImm + +� +k\S +Dm, Rn + + . +There is a natural map rImm(� +k\S Dm, Rn) → rConf(k \ S, Rn), which is the composition of +the natural map into rImm(� +k\S Dm, Rn), followed by evaluation at the centers of the discs. +By the main result of [AˇS22], this map is an equivalence. It follows that the connectivity of the +layers of HZ ∧ rImm(M, Rn) is determined by the connectivity of the total fiber of the cubical +diagram +S �→ HZ ∧ rConf(k \ S, Rn). +To analyze the total fiber of this cube, we need to review some facts about the homology of +r-configuration spaces. This will be done in the next section. +4. r-configuration spaces in Rn as complements of subspace arrangements +We saw in the previous section that the convergence of the Taylor tower of the functor HZ ∧ +rImm(−, Rn) is determined by the homology of r-configuration spaces rConf(k, Rn). These con- +figuration spaces can be interpreted as the complement of an arrangement of subspaces of (Rn)k. +The combinatorics and topology (in particular, homology and cohomology) of subspace arrange- +ments and their complements are well studied. Some of main references are [OS80], [GM80], +[GM83a], [GM83b], [BEZ90]. In particular, the (co)homology of r-configuration was studied from +13 + +this perspective first by Bj¨orner and Welker in [BW95], and by a number of people after that. +In this section we review a qualitative description of the cohomology of r-configuration spaces, +based on the Goresky-MacPherson formula. We will also describe the effect on cohomology of +restriction maps between configuration spaces. +Recall that an r-configuration space of k points in Rn is defined to be the space +rConf(k, Rn) = {(v1, ..., vk) ∈ (Rn)k : ∄1 ≤ i1 < · · · < ir ≤ k such that vi1 = ... = vir}. +The space rConf(k, Rn) is an example of the complement of a subspace arrangement. Let us +now recall some formal definitions. +Definition 4.1. Suppose I is an r-tuple of integers I = (i1, . . . , ir), where 1 ≤ i1 < · · · < ir ≤ k. +Let us denote the set of all such r-tuples by +�k +r +� +. Define +AI = {(v1, . . . , vk) ∈ (Rn)k | vi1 = · · · = vir}. +Let A = +� +AI | I ∈ +�k +r +�� +. When we need to make the set k explicit, we write Ak. More generally, +for any set T define AT to be the set of “r-equal” diagonals in (Rn)T. +Note that one can identify rConf(k, Rn) with the complement of the union of the AIs: +rConf(k, Rn) = (Rn)k \ +� +I∈(k +r) +AI. +Example 4.2. +• If k < r, rConf(k, Rn) ∼= (Rn)k ≃ ∗ +• If k = r, rConf(k, Rn) ∼= (Rn)r − ∆ ≃ S(r−1)n−1, where ∆ is the thin diagonal in (Rn)r +and S(r−1)n−1 is the sphere of dimension (r − 1)n − 1. +The collection A of linear subspaces of Rnk is an example of a subspace arrangement. Recall +that the intersection lattice of A is the poset LA consisting of all the intersections AI1 ∩· · ·∩AIt +of elements of A, ordered by reverse inclusion. We include in LA the “empty intersection” of +AIs, which is Rnk. The space Rnk is the minimal element of LA. It will be denoted by ˆ0. The +maximal element of LA is the intersection of all the AI, which, assuming k ≥ r, is the diagonal +copy of Rn in Rnk. We denote the maximal elements of LA by ˆ1. +The poset LA is isomorphic to the poset Πk,r of partitions of {1, . . . , k} whose every block is +either a singleton or contains at least r elements. We call elements of Πk,r r-equal partitions +of {1, . . . , k}. The partitions are ordered from finer to coarser. The isomorphism Πk,r → LA +sends a partition λ of {1, . . . , k} to the space of k-tuples of vectors (v1, . . . , vk) ∈ (Rn)k with +the property that vi = vj whenever i and j are in the same block of λ. Equivalently, one can +say that λ is sent to the space of functions from k to Rn that are constant on each block of λ. +From now on we will identify the posets LA and Πk,r. +Because LA is a partially ordered set, we can define the open interval (x, y) in LA to be the set +(x, y) = {z ∈ LA | x < z < y}. +Definition 4.3. The order complex ∆(x, y) of an open interval (x, y) in LA, is the abstract +simplicial complex whose vertices are the elements of (x, y) and whose p-simplices are the chains +x0 < ... < xp in (x, y). +14 + +Let �Hi(x, y) denote the ith reduced simplicial homology group of ∆(x, y) with integer coefficients. +Similarly, �H +i(x, y) denotes the ith reduced cohomology group of ∆(x, y). +The (reduced) cohomology groups of the space rConf(k, Rn) = Rnk\� +I∈(k +r) AI can be described +in terms of (reduced) homology groups of the order complex of intervals in the intersection lattice +of A. This is known as the Goresky-MacPherson formula. For the original proof of the Goresky- +MacPherson formula by means of stratified Morse theory see [GM88, Part III]. An elementary +proof was given by Ziegler and ˇZivaljevi´c in [ZˇZ93]. For the original calculation of the cohomology +rConf(k, Rn) using the Goresky-MacPherson formula see [BW95]. Here is the statement, in the +case relevant to us. +Theorem 4.4 (Special case of Goresky-MacPherson formula). There is an isomorphism +(10) +�H +i(rConf(k, Rn)) ∼= +� +x∈L>ˆ0 +A +�Hcodim(x)−2−i(ˆ0, x) +Here, the direct sum is indexed by all x ̸= ˆ0 in LA, and codim(x) is the codimension of the space +x as the subspace of Rnk. +For each diagonal x ∈ LA, let c(x) denote the number of components of the partition of k which +determines the diagonal x. Obviously, dimension of x in (Rn)k is dim(x) = n · c(x), so +(11) +codim(x) = n(k − c(x)). +The following easy example of 3-configuration spaces of 4 points illustrates the application of +formula (10). +Example 4.5. Let A is the set of all (at least 3)-diagonals in (Rn)4. Then 3 Conf(4, Rn) = +(Rn)4 − A. The intersection lattice LA of A is pictured in Figure 1. Using Theorem 4.4, we find +that for every n > 1, +H0(3 Conf(4, Rn)) ∼= Z, +H2n−1(3 Conf(4, Rn)) ∼= Z4, +H3n−2(3 Conf(4, Rn)) ∼= Z3, +and other cohomology groups are trivial. For n = 1, the formula is still valid, except that in this +case 2n−1 = 3n−2 = 1, so the two cohomology groups add together. So H0(3 Conf(4, R)) ∼= Z +and H1(3 Conf(4, R)) ∼= Z7. For n = 1, 2, the cohomology of 3 Conf(4, Rn) can be read off the +tables at the end of [BW95]. +For the purpose of analysing the layers in the homological Taylor tower for r-immersions it also is +desirable to know the effect of restriction maps between r-configuration spaces on cohomology. +Suppose we have a subset T ⊂ {1, . . . , k}. Then we have a restriction map rConf(k, Rn) → +rConf(T, Rn). We want to describe the induced homomorphism on cohomology, in terms of +formula (10). The inclusion T ֒→ {1, . . . , k} induces an inclusion of the poset of r-equal partitions +of T into the poset of r-equal partitions of {1, . . . , k}, by making each element of {1, . . . , k} \ T +into a singleton. Notice that for every r-equal partition of T, the codimension of the corresponding +diagonal is the same whether it is considered a diagonal in (Rn)T or in (Rn)k. This is so because +the codimension of a diagonal determined by a partition is determined by the difference between +15 + +(1)(2)(3)(4) +(1)(2, 3, 4) +(1, 2, 3, 4) +(2)(1, 2, 3) +(3)(1, 2, 4) +(4)(1, 2, 3) +Figure 1. Intersection lattice for 3 Conf(4, Rn), also known as Π4,3 +the cardinality of the set and the number of blocks of the partition, by formula (11). This number +remains unchanged if one adds some singletons to a partition. Thus we have a homomorphism +(12) +� +x∈L>ˆ0 +AT +�Hcodim(x)−2−i(ˆ0, x) → +� +x∈L>ˆ0 +Ak +�Hcodim(x)−2−i(ˆ0, x) +which is defined by the inclusion L>ˆ0 +AT ֒→ L>ˆ0 +Ak, and uses the fact that for every x ∈ L>ˆ0 +AT , the +number codim(x) is the same whether x is considered an element of L>ˆ0 +AT or of L>ˆ0 +Ak. +Lemma 4.6. The homomorphism �H +i(rConf(T, Rn)) → �H +i(rConf(k, Rn)) corresponds, under +the isomorphism (10), to the homomorphism (12) that we just described. +Proof. This follows easily from the fact that the Goresky-MacPherson formula is natural with +respect to inclusions of subarrangements [Hu94, Corollary 2.1] +□ +5. Total fiber of a retractive cubical diagram +In general homotopy groups do not commute with total homotopy fibers of cubical diagrams. +In this section we will show that for a class of cubes that we call retractive they do commute. +More precisely, we show that for retractive cubes, the homotopy groups of the total fiber are +canonically isomorphic to the total kernel of the cube of homotopy groups. +Suppose we have a two-dimensional cubical diagram of spaces or spectra +(13) +E∅ +i∅,1 � +i∅,2 +� +E1 +i1,12 +� +E2 +i2,12 +� E12 +16 + +Suppose that all the maps in the square (13) have homotopy sections, so that the square of +sections +E12 +s12,1 � +s12,2 +� +E1 +s1,∅ +� +E2 +s2,∅ +� E0 +commutes up to homotopy, and so that the following mixed square +E2 +i2,12 � +s2,∅ +� +E12 +s12,1 +� +E0 +i∅,1 +� E1 +also commutes up to homotopy. Note that the vertical maps in the mixed square are sections, +while the horizontal maps are from the original square. +Let us call a square (13) with such sections a retractive square. +More generally, let us define a retractive cubical diagram as follows. +Definition 5.1. Let χ be a k-dimensional cubical diagram. We say that χ is retractive if for +every U ⊂ {1, . . . , k} and every i /∈ U, the map χ(U) → χ(U ∪ {i}) has a homotopy section, +the cube of sections commutes up to homotopy, and furthermore whenever U ⊂ {1, . . . , k}, and +i, j ∈ {1, . . . , k} \ U, with i < j, the following mixed square commutes up to homotopy +χ(U ∪ {j}) +� +� +χ(U ∪ {i, j}) +� +χ(U) +� χ(U ∪ {i}) +. +Lemma 5.2. Let χ be a retractive k-dimensional cubical diagram of spectra. Let E∗ be any +homology theory, and let E∗ be a cohomology theory. Then E∗(tfiber χ) (resp. E∗(tfiber χ)) is a +direct summand of E∗(χ(∅)) (resp. of E∗(χ(∅))). Moreover, the following natural homomorphism +is an isomorphism: +E∗(tfiber χ) +∼ +=−→ tkernel (E∗χ) . +Similarly, there is a natural isomorphism +tcokernel(E∗χ) +∼ +=−→ E∗(tfiber χ). +Proof. We will prove the claim for homology. The proof of the cohomological statement is the +same, reversing all arrows. The proof is by induction on k, starting with with the case k = 1, +which is elementary and well-known. Let us review it anyway. A retractive 1-dimensional cube +is a map χ(∅) → χ(1), together with a homotopy section χ(1) → χ(∅). The total fiber of +the cube is the homotopy fiber of the map χ(∅) → χ(1). By homotopy section we mean that +the composition χ(1) → χ(∅) → χ(1) is a weak equivalence. It follows that the composition +E∗χ(1) → E∗χ(∅) → E∗χ(1) is an isomorphism. From here it readily follows that the long +exact sequence in E∗ associated with the fibration sequence tfiber χ → χ(∅) → χ(1) splits as a +17 + +direct sum of split short exact sequences in each degree. Furthermore it readily follows that the +following homomorphisms are isomorphisms +E∗ tfiber χ +∼ +=−→ ker (E∗χ(∅) → E∗χ(1)) +∼ +=−→ coker (E∗χ(1) → E∗(χ(∅))) . +Now suppose the lemma holds for cubes of dimension less than k and let χ be a retractive +cube of dimension k. Let χ1 and χ2 be k − 1-dimensional cubes defined by χ1(U) = χ(U) +and χ2(U) = χ(U ∪ {k}). Then χ can be identified with the natural map of cubes χ1 → χ2. +The cubes χ1 and χ2 are retractive, so by induction hypothesis, the lemma holds for them. The +retractions do not quite define a map of cubes χ2 → χ1, because we only assumed that the mixed +squares commute up to homotopy. But they do define a homomorphism of cubes E∗χ2 → E∗χ1, +which is a section of the homomorphism of cubes E∗χ1 → E∗χ2. We have the following diagram +E∗ tfiber χ +E∗ tfiber χ1 +E∗ tfiber χ2 +tkernel E∗χ2 +tkernel E∗χ1 +tkernel E∗χ2 +∼ += +∼ += +∼ += +The top row is induced by applying E∗ to a fibration sequence of spectra. The vertical homomor- +phisms are isomorphisms by induction hypothesis. It follows that the upper right homomorphism +is a split surjection, and the top row is a split short exact sequence in each dimension. Fur- +thermore, E∗ tfiber χ maps isomorphically onto the kernel of the bottom right map, which is +tkernel E∗χ. +□ +6. The cube of r-configuration spaces is retractive +Lemma 6.1. The k-cube of spaces +S �→ rConf(k \ S, Rn) +is retractive for n ≥ 2. +Proof. Let T be a finite set and suppose x /∈ T. Our first step it to construct a section to the +restriction map +rT∪{x},T : rConf(T ∪ {x}, Rn) → rConf(T, Rn). +Let p1: Rn → R be projection onto the first coordinate. Define a map +sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn) +as follows. An element of rConf(T, Rn) is a function f : T → Rn with the property that no r +points of T go to the same point. Extend f to a function from T ∪ {x} by sending x to +(max{p1f(t) | t ∈ T} + 1, 0, . . . , 0). +In words, x is sent to the point of Rn whose first coordinate is one more than the maximal +first coordinate of the existing points, and all other coordinates are zero. It is clear that the +image of x is different from all the other points in the configuration. +Thus if f was an r- +immersion, then the resulting map T ∪ {x} → Rn is still an r-immersion. We have defined a +18 + +map sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn). It is clear that the following composition +is the identity (not even just homotopic to the identity but is the actual identity map) +rConf(T, Rn) +sT,T ∪{x} +−−−−−→ rConf(T ∪ {x}, Rn) +rT ∪{x},T +−−−−−→ rConf(T, Rn). +It follows that sT,T∪{x} is a section of rT∪{x},T. Next, we need to show that whenever x, y /∈ T, +the following diagram commutes up to homotopy +rConf(T, Rn) +� +� +rConf(T ∪ {x}, Rn) +� +rConf(T ∪ {y}, Rn) +� rConf(T ∪ {x, y}, Rn) +It is for this step that we need to assume n ≥ 2. +Let f : T → Rn represent an element +of rConf(T, Rn). The images of f in rConf(T ∪ {x, y}, Rn) under the two ways around the +diagram are two extensions of f from T to T ∪ {x, y}. +One of the extensions sends x to +(max{p1f(t) | t ∈ T} + 1, 0, . . . , 0), and sends y to (max{p1f(t) | t ∈ T} + 2, 0, . . . , 0). The +other extension does the same thing, with x and y switched. It is clear that one can write a +homotopy between the two maps, by swapping the images of x and y along a circle in the plane +spanned by the first two coordinates of Rn. +Finally we need to check that the following mixed square commutes up to homotopy +rConf(T ∪ {x}, Rn) +� +� +rConf(T, Rn) +� +rConf(T ∪ {x, y}, Rn) +� rConf(T ∪ {y}, Rn) +. +This, too, is clear. In fact, it is easy to check that there is a well-defined straight line homotopy +between the two maps around the square. +We have shown that the section maps that we have defined make the cube of r-configuration +spaces and restriction maps between them into a retractive cube. +□ +7. Connectivity of the cube of (co)homologies of r-configuration spaces +We have seen that the cube of spaces S �→ rConf(k \ S, Rn), where S ranges over the subsets +of {1, . . . , k} is retractive (Lemma 6.1). It follows that the cube of spectra obtained by applying +the suspension spectrum functor to it, i.e., the cube +(14) +S �→ Σ∞ rConf(k \ S, Rn), +is also retractive. +Our goal is to analyse how cartesian is the cube S �→ HZ∧Σ∞ rConf(k\S, Rn). Smash product +commutes with total fibers of cubical diagrams of spectra. Therefore, the answer is the same +as for the cubical diagram (14). However, we want to use the description of the cohomology +of r-configuration spaces given by the Goresky-MacPherson formula. The following lemma says +that the homology and cohomology groups of the relevant spectrum are isomorphic. +Lemma 7.1. The homology and cohomology groups of the total fiber of (14) are (non-canonically) +isomorphic. +19 + +Proof. It is known, for example by the results of [BW95], that the homology groups of the space +rConf(k, Rn), and therefore also of the suspension spectrum of this space, are finitely generated +free abelian groups. Since the cube Σ∞ rConf(k \ S, Rn) is retractive, it follows by Lemma 5.2 +that the homology of the total fiber of the cube Σ∞ rConf(k \ S, Rn) is a direct summand of +the homology of Σ∞ rConf(k, Rn). Therefore, the homology groups of the total fiber are also +finitely generated free abelian groups. Therefore they are isomorphic to the cohomology groups +of the total fiber, by the universal coefficients theorem. +□ +It follows that the homological connectivity of the total fiber of (14) is equivalent to the coho- +mological connectivity. Next, we give a qualitative description of the cohomology of the total +fiber, in the style of Theorem 4.4. +Let Π≥r(k) denote the set partitions of k with the property that each component has at least r +elements (i.e., elements of Πk,r without singletons). +Lemma 7.2. The i-th cohomology group of the total fiber of the cube (14) is isomorphic to the +following direct sum: +(15) +� +x∈Π≥r(k) +�Hcodim (x)−2−i(ˆ0, x) +Proof. The cube (14) is retractive. Using the cohomological part of Lemma 5.2, we conclude +that the i-th cohomology of the total fiber is isomorphic to the cokernel of the homomorphism +k +� +i=1 +�H +i rConf(k \ {i}, Rn) → �H +i rConf(k, Rn). +By Lemma 4.6, this homomorphism can be identified with the following homomorphism +(16) +k +� +i=1 +� +x∈L>ˆ0 +Ak\{i} +�Hcodim(x)−2−i(ˆ0, x) → +� +x∈L>ˆ0 +Ak +�Hcodim(x)−2−i(ˆ0, x) +The homomorphism maps each summand in the source isomorphically onto a summand in the tar- +get (some summands in the source go to the same summand in the target, so the homomorphism +is not injective). The image of the homomorphism is the sum of terms corresponding to r-equal +partitions with at least one singleton. The cokernel is the direct sum of terms corresponding to +r-equal partitions that do not have a singleton. +□ +It follows from Lemma 7.2 that to find how cartesian the cube (14) is, we need to find the +smallest i for which the homology group +(17) +�Hcodim(x)−2−i(ˆ0, x) +is non-trivial for some x ∈ Π≥r(k). +Throughout this section, let x be a partition of {1, . . . , k} where each block has at least r +elements. Recall that c(x) denotes the number of blocks of x. Note that if k1, . . . , kc(x) are the +sizes of the blocks of x, then k1 + · · · + kc(x) = k. Let [ˆ0, x] be the closed interval in Πk,r. +Lemma 7.3. Let x be as above. Suppose x has c(x) blocks, of sizes k1, . . . , kc(x). Then there +is an isomorphism of posets +[ˆ0, x] ∼= Πk1,r × · · · × Πkc(x),r. +20 + +Proof. The interval [ˆ0, x] consists of r-equal partitions of {1, . . . , k} that are refinements of x. +This is the same data as an r-equal partition of each block of x, which is the same as an element +of Πk1,r × · · · × Πkc(x),r. +□ +Given a poset P with a minimum and maximum element, let P0 be the poset P with the minimum +and maximum removed. +Corollary 7.4. Let x be as in the previous lemma. Then there is a homotopy equivalence (∗ +denotes joint) +|∆(ˆ0, x)| ≃ Σc(x)−1|Π0 +k1,r| ∗ · · · ∗ |Π0 +kc(x),r|. +Proof. This follows from the lemma, and the well-known fact that given two posets P and Q +with minimum and maximum objects, there is a homotopy equivalence [Wal88, Theorem 5.1 (d)] +|(P × Q)0| ≃ Σ|P0| ∗ |Q0|. +□ +Lemma 7.5. Let x be as in the previous lemma and corollary. Then |∆(ˆ0, x)| is homotopy +equivalent to a complex of dimension k−c(x)(r−1)−2. Furthermore, the homology of |∆(ˆ0, x)| +in dimension k − c(x)(r − 1) − 2 is non zero. +Proof. By the corollary, the space |∆(ˆ0, x)| is homotopy equivalent to Σc(x)−1|Π0 +k1,r| ∗ · · · ∗ +|Π0 +kc(x),r|. By the results of [BW95], |Π0 +k,r| is homotopy equivalent to a wedge of spheres, not all +of the same dimension, and the top homology of this space occurs in dimension k − r − 1. It +follows that the space Σc(x)−1|Π0 +k1,r|∗· · ·∗|Π0 +kc(x),r| is a wedge of spheres, with the top homology +occurring in dimension +c(x) − 1 + (k1 − r − 1) + · · · + (kc(x) − r − 1) + c(x) − 1 = k − c(x)(r − 1) − 2. +□ +Example 7.6. If r ≤ k < 2r, there is only one summand x in (15) - this is the partition {k}, or +in other words the thin diagonal. For this x, dim ∆(ˆ0, x) = k − r − 1. +Now we can state and prove the main result of this section +Proposition 7.7. When r ≤ n + 1, the cube (14) is k(n − 1) + +� k +r +� +(r − n − 1)-cartesian. +When r ≥ n + 1, the cube (14) is k(n − 1) + r − n − 1-cartesian. +Remark 7.8. Note that when r = n+1 both formulas say that the cube (14) is k(n−1)-cartesian. +Proof. Given x, the smallest i for which the homology (17) might be non-trivial is one that +satisfies codim(x)−2−i = dim ∆(ˆ0, x). Using Lemma 7.5 we have that the smallest i for which +the total cokernel (15) might be non-trivial is one that satisfies +codim(x) − 2 − i = k − c(x)(r − 1) − 2. +Because codim(x) = n(k − c(x)) for x ∈ Π≥r(k), it follows that +(18) +i = k(n − 1) + c(x)(r − n − 1). +21 + +We have to see for which x this number i is the smallest possible. We distinguish between two +overlapping cases, depending on the sign of r − n − 1. +1) When r − n − 1 ≤ 0, i.e. when r ≤ n + 1, finding i as small as possible is the same as finding +x ∈ Π≥r(k) with the biggest number c(x) of components. Since all components have to be of +the size at least r, the largest number of them is attained when there is a maximum number of +them of the size r. In that case, c(x) = +� k +r +� +, so the smallest i is +i = k(n − 1) + +�k +r +� +(r − n − 1). +So in this case, the cubical diagram (14) is k(n − 1) + +� k +r +� +(r − n − 1)-cartesian. +2) When r − n − 1 ≥ 0, i.e. r ≥ n + 1, finding i as small as possible is the same as finding +x ∈ Π≥r(k) with the smallest number c(x) of components. Thus we need c(x) to be equal to 1. +This x is actually the thin diagonal in the space (Rn)k that corresponds to the partition {k} of +k. In that case, +i = k(n − 1) + r − n − 1, +hence (14) is k(n − 1) + r − n − 1-cartesian. +□ +8. Convergence result +Let M be a smooth manifold of dimension m. Now we finally can calculate the connectivity of +the map +(19) +TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn). +Knowing that ck-connectivity of the total fiber of the cube (14) implies (ck−km+1)-connectivity +of the map (19), we can find the conditions under which the Taylor tower converges, using results +from Section 7. There are three different cases. +1) For r − n − 1 < 0, the connectivity of the map (19) is +(20) +k(n − 1) + +�k +r +� +(r − n − 1) − 1 − mk + 1 = k(n − m − 1) + +�k +r +� +(r − n − 1) += k(n − m − 1) + +�k +r − k mod r +r +� +(r − n − 1) += k +� +n − m − n +r − 1 +r +� +− k mod r +r +(r − n − 1) += k +� +nr − 1 +r +− m − 1 +r +� +− k mod r +r +(r − n − 1) +where we noted that +� k +r +� += k/r − (k mod r)/r. Note now that +−k mod r +r +(r − n − 1) +22 + +is nonnegative since r − n − 1 < 0. This means that, as long as +nr − 1 +r +− m − 1 +r > 0, +the connectivities increase with k. +2) For r − n − 1 = 0, the connectivity of the map (19) is +(21) +k(n − 1) − 1 − mk + 1 = k(n − m − 1), +which goes to +∞ as k −→ +∞ if n − m − 1 > 0. +3) For r − n − 1 > 0, the connectivity of the map (19) is +(22) +k(n − 1) + r − n − 2 − mk + 1 = k(n − m − 1) + r − n − 1, +which goes to +∞ as k −→ +∞ if n − m − 1 > 0. +Thus we proved the following theorem. +Theorem 8.1. (Homological convergence of the Taylor tower for r-immersions in Rn) +Let M be an m-dimensional smooth manifold and Rn the n-dimensional Euclidean space. Assume +n > 1. Let rImm(M, Rn) be the space of r-immersions of M in Rn. Consider the map +pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn). +a) For r ≤ n + 1 the map pk is +k +� +nr − 1 +r +− m − 1 +r +� +− k mod r +r +(r − n − 1) +-connected. The tower converges intrinsically if n > rm+1 +r−1 . +b) For r ≥ n + 1 the map pk is k(n − m − 1) + r − n − 1-connected. The tower converges +intrinsically if n > m + 1. +Proof. Only the assertions regarding intrinsic convergence remain to be checked. +The tower +converges intrinsically if the connectivity of pk approaches ∞ with k. In the case r ≤ n+1, since +(k mod r) is a bounded function of k, this is equivalent to the condition nr−1 +r +− m − 1 +r > 0, +which is the same as n > rm+1 +r−1 . In the case r ≥ n + 1, the formula for the connectivity of pk +clearly tells us that the connectivity goes to ∞ if n > m + 1. +□ +9. Comparing with the unstable tower +In this section we will compare the layers, and the connectivities of the maps in the Taylor tower +of HZ ∧ rImm(M, Rn) with those in the Taylor tower of the unstabilized functor rImm(M, Rn). +We will show that roughly up to degree 2r − 1 the connectivities of the maps in the two towers +are the same, and the first non-trivial homotopy groups of the layers are isomorphic. +23 + +In this section, let us assume that we chose a basepoint in rImm(M, Rn) rather than just in +Imm(M, Rn), so that the presheaf rImm(−, Rn) takes values in pointed spaces. We have a +diagram of presheaves +(23) +rImm(−, Rn) +i←− rImm(−, Rn) +s−→ Ω∞Σ∞rImm(−, Rn) +h−→ Ω∞HZ ∧ rImm(−, Rn). +The map i induces an equivalence of derivatives and layers beyond the first one. The map h +induces the Hurewicz homomorphism. In particular, it induces a Hurewicz isomorphism on the +first non-trivial homotopy group of each layer. We focus on the question for which k the map s, +and therefore also h ◦ s, induces an isomorphism on the first nontrivial homotopy group of the +k-th layer. When r = 2, the answer is known to be: only for k = 2. We show that for r > 2 the +answer is: for all k ≤ 2r − 1, with a small caveat for r = 3. +Theorem 9.1. Assume 0 < dim(M) < n, r > 2. +For 1 < k < r, the following maps are equivalences: +Tk rImm(M, Rn) → T1 rImm(M, Rn) ≃ Imm(M, Rn) +and +TkHZ ∧ rImm(M, Rn) → T1HZ ∧ rImm(M, Rn) ≃ ∗. +For r ≤ k ≤ 2r − 1, the connectivity of the map Tk rImm(M, Rn) → Tk−1 rImm(M, Rn) is the +same as the connectivity of the map TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn). +When r = 3, k = 2r − 1 = 5, the map s, and therefore also h ◦ s in diagram (23), induces an +epimorphism on the first non-trivial homotopy group of the k-th layer. In all other cases when +r > 2, r ≤ k ≤ 2r − 1, the maps s and h ◦ s induce an isomorphism on the first non-trivial +homotopy group of the k-th layer. +Remark 9.2. The case k = r, r + 1 of the last assertion of the theorem can be obtained by +comparing our Theorem 8.1 with the calculations done in [SˇSV20]. +Proof. The assertion that T1 rImm(M, Rn) ≃ Imm(M, Rn) follows from the fact that when +M = Dm, the following maps are equivalences [AˇS22] +Emb(Dm, Rn) +≃−→ rImm(Dm, Rn) +≃−→ Imm(Dm, Rn), +together with the fact that the functor Imm(−, Rn) is linear, at least on manifolds whose handle +dimension is less than n. +The assertion that both towers are constant for k < r follows from the fact that the derivatives of +both functors vanish below degree r. Indeed, the k-th layer in the Taylor tower of rImm(M, Rn) +is determined by the following k-dimensional cubical diagram +S �→ rImm( +� +k\S +Dm, Rm). +By the result of [AˇS22], this cubical diagram is equivalent to the diagram +S �→ L(Rm, Rn)k\S × rConf(k \ S, Rn). +24 + +Here L(Rm, Rn) is the space of injective linear maps from Rm to Rn. This is the “tangential +data” of an immersion. When k > 1 the tangential data cancels out, and the last cube is as +cartesian as the following cube +(24) +S �→ rConf(k \ S, Rn). +On the other hand, the k-th layer in the Taylor tower of Ω∞Σ∞ rImm(M, Rn) is determined by +the following k-dimensional cubical diagram +(25) +S �→ Ω∞Σ∞ rConf(k \ S, Rn). +To prove the assertion about the connectivities of the maps in the two towers, we need to +show that the cubical diagram (24) is as cartesian as the diagram (25) in the indicated cases. +Furthermore, we want to prove that the map s in (23) induces an isomorphism/epimorphism on +the first non-trivial homotopy groups of the total homotopy fibers in the appropriate cases. +The map s induces the following map of cubical diagrams, indexed by the poset of subsets S ⊂ k, +(26) +rConf(k \ S, Rn) +s−→ Ω∞Σ∞ rConf(k \ S, Rn). +The spaces rConf(k \ S, Rn) are (r − 1)n − 2-connected. By Freudenthal suspension theorem, +the maps (26) are 2(r−1)n−3-connected. On the other hand, both cubes are retractive cubes by +Lemma 6.1. It follows that the homotopy groups of the total homotopy fibers of both cubes are +isomorphic to the total kernels of the corresponding cubes of homotopy groups. Proposition 7.7 +tells us the connectivity of the total homotopy fiber of (25), and therefore also the connectivity +of the total kernel of the corresponding cube of homotopy groups. If this connectivity is smaller +than (resp. equals to) the connectivity of the maps in (26), then (26) induces an isomorphism +(resp: an epimorphism) between the first non-trivial homotopy groups of the total homotopy +fibers. So we have to check that the range provided by Proposition 7.7 is smaller than (or equals +to) 2(r − 1)n − 3 in the cases indicated in the statement that we are trying to prove. +Suppose first that r > n+ 1. In this case, Proposition 7.7 says that (25) is k(n−1) + r −n−1- +cartesian. So we have to check that the inequality +k(n − 1) + r − n − 1 < 2(r − 1)n − 3 +holds whenever k < 2r. Simplifying, we obtain the inequality +k < (2n − 1)r − 3 +n − 1 +− 1. +So it is enough to check the inequality +2r ≤ (2n − 1)r − 3 +n − 1 +− 1. +Multiplying by n − 1 we obtain the inequality +2r(n − 1) ≤ (2n − 1)r − 3 − n + 1, +which is equivalent to r ≥ n + 2, which is what we assumed. +Now suppose that r ≤ n + 1. Then Proposition 7.7 says that (25) is k(n − 1) + +� k +r +� +(r − n − 1)- +cartesian. So we have to check that the inequality +k(n − 1) + +�k +r +� +(r − n − 1) < 2(r − 1)n − 3 +25 + +holds when r ≤ k < 2r, with the exception that when r = 3, k = 5 it is in fact an equality. The +reader can check that in this case we do indeed obtain the equality +5(n − 1) + +�5 +3 +� +(3 − n − 1) = 4n − 3. +In other cases, the assumption r ≤ k < 2r implies +� k +r +� += 1. So we have to check the inequality +k(n − 1) + r − n − 1 < 2(r − 1)n − 3. +We can rewrite the inequality as follows +k(n − 1) < (2r − 1)(n − 1) + r − 3, +or equivalently +k < 2r − 1 + r − 3 +n − 1. +For r = 3 this inequality is equivalent to k < 5. For 3 < r ≤ n + 1, this holds for all k ≤ 2r − 1, +as stated. +□ +10. Further questions +1. We gave conditions on the m and n that guarantee intrinsic convergence of the Taylor tower +of HZ ∧ rImm(M, Rn). The next question is, what does the Taylor tower from Theorem 8.1 +converge to? It is natural to guess that whenever the Taylor tower converges intrinsically, it +actually converges to HZ ∧ rImm(M, Rn). +2. +What can one say about the convergence of the Taylor tower for the unstable functor +rImm(M, Rn)? The question of intrinsic convergence of the unstable tower might be tractable, +and is a good place to start. One can use the methods of this paper to describe the layers +of the functor Σ∞rImm(M, Rn). Given this, one can try to analyse the layers of the functor +rImm(M, Rn) via the cobar construction +cobar(Ω∞, Σ∞Ω∞, Σ∞rImm(M, Rn)), +in the style of [AC11]. It is conceivable that one can use these methods to obtain conditions on +m, n, and r that guarantee that the tower converges intrinsically. +Then there is a question of what the tower actually converges to. Once again, it seems reasonable +to guess that whenever the Taylor tower of a “natural” functor converges intrinsically, then it +actually converges to the functor. +3. What can one say about r-immersions into a general manifold N? In order to understand the +layers of the tower of the functor Σ∞rImm(M, N) one needs to understand (the stable homotopy +type of) the homotopy fiber of the map rConf(k, N) → Nk. For r = 2 this homotopy fiber was +analysed in [Aro09], and it seems likely that a similar analysis can be done for general r. +4. Construct interesting invariants/obstructions to existence of r-immersions, using the Taylor +tower. In this paper we focused on situations where the connectivity of the k-th layer in the +tower goes to infinity as k goes to infinity. But situations when the connectivity does not go to +infinity also can be interesting. Of particular potential interest are situations where the layers are +all either −1-connected or −2-connected. In the former case, the bottom homotopy groups of +the layers give invariants, in the latter case they give obstructions to existence. +26 + +For example, it follows from Theorem 8.1 that when n = m+1 and r = n+1, then all the layers +of HZ∧(n+ 1) Imm(M, Rn) are −1-connected. The 0-th homotopy groups of the layers should +give invariants of r-immersions. In the case n = 2, and say M = S1, 3 Imm(S1, R2) is the space +of smooth curves in R2 that do not have triple intersections. Spaces of such curves were studied +quite intensely, starting with Arnol’d [Arn94, Tab96, Shu95]. In particular, Arnol’d developed the +theory of finite type invariants for such curves. We expect these invariants to show up in the +Taylor tower of HZ ∧ 3 Imm(S1, R2). In particular we speculate that the first non-trivial layer of +the tower, which by Theorem 9.1 is the third layer, detects the “Strangeness” invariant, defined +in [Arn94] and studied further in [Tab96] and [Shu95]. +References +[AC11] +Greg Arone and Michael Ching. Operads and chain rules for the calculus of functors. Ast´erisque +338:vi+158, 2011. +[ALV07] +Gregory Arone, Pascal Lambrechts, and Ismar Voli´c. Calculus of functors, operad formality, and rational +homology of embedding spaces. Acta Math. 1999(2):153–198, 2007. +[Arn94] +V. I. Arnol’d. Plane curves, their invariants, perestroikas and classifications. In Singularities and bi- +furcations, volume 21 of Adv. Soviet Math., pages 33–91. Amer. Math. 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Topol., +4(2):383–405, 2011. +[MV15] +Brian A. Munson and Ismar Voli´c. Cubical homotopy theory, volume 25 of New Mathematical Mono- +graphs. Cambridge University Press, Cambridge, 2015. +[OS80] +Peter Orlik and Louis Solomon. Combinatorics and topology of complements of hyperplanes. Invent. +Math., 56(2):167–189, 1980. +[Shu95] +Alexander Shumakovich. Explicit formulas for the strangeness of plane curves. Algebra i Analiz, +7(3):165–199, 1995. +[SˇSV20] +Bridget Schreiner, Franjo ˇSarˇcevi´c, and Ismar Voli´c. Low stages of the Taylor tower for r-immersions +Involve, a J. of Math., 13(1):51–75, 2020. +[ST16] +Paul Arnaud Songhafouo Tsopm´en´e. The rational homology of spaces of long links. Algebr. Geom. +Topol., 16(2):757–782, 2016. +[Tab96] +Serge Tabachnikov. Invariants of smooth triple point free plane curves. J. Knot Theory Ramif., +5(4):531–552, 1996. +[Vol06] +Ismar Voli´c. Finite type knot invariants and the calculus of functors. Compos. 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Ann., 295:527–548, 1993. +Gregory Arone +Department of Mathematics, Stockholm University +Email address: gregory.arone@math.su.se +Franjo ˇSarˇcevi´c +Department of Mathematics, University of Sarajevo +Email address: franjo.sarcevic@live.de +URL: pmf.unsa.ba/franjos +28 + diff --git a/INE1T4oBgHgl3EQfXwS1/content/tmp_files/load_file.txt b/INE1T4oBgHgl3EQfXwS1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4e19bf316e0623299d2cba7628390a6e3f6d475 --- /dev/null +++ b/INE1T4oBgHgl3EQfXwS1/content/tmp_files/load_file.txt @@ -0,0 +1,1057 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf,len=1056 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='03131v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='AT] 9 Jan 2023 INTRINSIC CONVERGENCE OF THE HOMOLOGICAL TAYLOR TOWER FOR r-IMMERSIONS IN Rn GREGORY ARONE AND FRANJO ˇSARˇCEVI´C Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For an integer r ≥ 2, the space of r-immersions of M in Rn is defined to be the space of immersions of M in Rn such that at most r − 1 points of M are mapped to the same point in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space of r-immersions lies “between” the embeddings and the immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We calculate the connectivity of the layers in the homological Taylor tower for the space of r- immersions in Rn (modulo immersions), and give conditions that guarantee that the connectivity of the maps in the tower approaches infinity as one goes up the tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We also compare the homological tower with the homotopical tower, and show that up to degree 2r − 1 there is a “Hurewicz isomorphism” between the first non-trivial homotopy groups of the layers of the two towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Prerequisites 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Cubical diagrams 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Manifold calculus of functors 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Spectra 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homological Taylor tower for reduced r-immersions in Rn 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' r-configuration spaces in Rn as complements of subspace arrangements 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Total fiber of a retractive cubical diagram 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The cube of r-configuration spaces is retractive 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Connectivity of the cube of (co)homologies of r-configuration spaces 19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Convergence result 22 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Comparing with the unstable tower 23 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Further questions 26 References 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Introduction Let M be a smooth manifold of dimension m, and fix an integer r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' An r-immersion of M in Rn is an immersion of M in Rn such that the preimage of every point in Rn contains at most r − 1 points of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space of r-immersions of M in Rn is denoted by rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Primary: 57R42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Secondary: 55R80, 57R40, 55P42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' calculus of functors, manifold calculus, Taylor tower, embeddings, immersions, r- immersions, homotopy of spectra, homological convergence, partial configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' ˇSarˇcevi´c was partially supported by the grant P20 01109 (JUNTA/FEDER, UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 1 r = 2, 2-immersions are the same thing as injective immersions, which are essentially the same as embeddings in nice cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In any case, we have inclusions of subspaces Emb(M, Rn) ⊆ 2 Imm(M, Rn) ⊂ 3 Imm(M, Rn) ⊂ · · · ⊂ rImm(M, Rn) ⊂ · · · ⊂ Imm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this paper we study the homological Taylor tower of the r-immersions functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The “Taylor tower” is meant in the sense of manifold calculus (also known as embedding calculus) developed by Weiss [Wei99] and Goodwillie-Weiss [GW99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The basic idea of manifold calculus is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In order to study the homotopy type of a space such as rImm(M, Rn), one views it as a particular value of the presheaf rImm(−, Rn) defined on M (one can also consider more general target manifolds than Rn, but we will content ourselves with maps into Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A presheaf is a contravariant functor on the poset O(M) of open subsets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Inside O(M) there is a sequence of subposets O1(M) ⊂ · · ·Ok(M) ⊂ · · · ⊂ O∞(M), where Ok(M) is the poset of open subsets of M that are diffeomorphic to the disjoint union of at most k copies of Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By restricting a presheaf F to Ok(M) and then extrapolating back to O(M) one obtains a tower of approximations to F, which is usually denoted as follows F → (T∞F → · · · → TkF → Tk−1F → · · · T0F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is called the “Taylor tower” of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Manifold calculus, and the Taylor tower in particular, has had many consequences and applications [Mun05], [Vol06], [ALV07], [Mun11], [DH12], [ST16], [BdBW18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this paper we investigate the Taylor tower that calculates the homology of the space rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In practice, this means the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' First of all, it is convenient to replace the space of r- immersions with r-immersions modulo immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us suppose that we fix a basepoint in Imm(M, Rn), and let rImm(M, Rn) be the homotopy fiber of the inclusion map rImm(M, Rn) → Imm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let HZ denote the Eilenberg-MacLane spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We are interested in the Taylor tower of the presheaf of Spectra, defined by the formula U �→ HZ ∧ rImm(U, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (more precise definitions are given in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Our main result concerns the rate of convergence of the Taylor tower of this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The question of convergence is a fundamental one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will distinguish between two aspects of convergence: how strongly the tower converges to its limit, and what it converges to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will say that the Taylor tower of a functor F converges intrinsically at M if the connectivity of the map TkF(M) → Tk−1F(M) approaches ∞ as k approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We say that the Taylor tower of F converges strongly to F(M) if the connectivity of the map F(M) → TkF(M) approaches ∞ as k approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Strong convergence implies intrinsic convergence, but the converse does not have to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In practice it seems that for “natural” functors that we know, whenever the Taylor tower of F converges intrinsically, it converges strongly to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' But intrinsic convergence is usually much easier to prove than strong convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Before we state our main result, let us recall, for context, that one of the deepest results in functor calculus is the Goodwillie-Klein-Weiss convergence theorem [GW99], [GK08], [GK15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 (Convergence of the Taylor tower for spaces of embeddings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If M is a smooth closed manifold of dimension m, and N is a smooth manifold of dimension n, then the map Emb(M, N) → Tk Emb(M, N) 2 is k(n − m − 2) + 1 − m-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, if n−m−2 > 0, then the connectivities grow with k and the Taylor tower therefore converges strongly to Emb(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There is an easier, but also important convergence result for the homological version of the tower, which is more directly relevant to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define Emb(M, Rn) to be the homotopy fiber of the inclusion Emb(M, Rn) → Imm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Consider the contravariant functor from O(M) to Spectra that sends U to HZ ∧ Emb(U, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This functor represents the homology of the space of embeddings modulo immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The Taylor tower of this functor is known to converge when n > 2m + 1 [Wei04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Now let us state our main result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let M be m-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Assume that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If r ≤ n + 1, the Taylor tower for HZ ∧ rImm(M, Rn) converges intrinsically when n > rm + 1 r − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If r ≥ n + 1 then the Taylor tower converges intrinsically when n > m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (1) When r = n + 1 the two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Indeed, the function f(n) = n2 − nm − m − 1, n ∈ N, is positive only for n > m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (2) When r = 2 we get the condition n > 2m + 1, which is the known condition for the convergence of the Taylor tower of HZ ∧ Emb(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (3) The condition n > rm+1 r−1 is equivalent to rm−(r−1)n < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The number rm−(r−1)n equals, at least when it is positive, to the dimension of the intersection of r copies of Rm embedded in Rn in a general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Next let us discuss the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let F be a presheaf defined on a suitable category of m-dimensional manifolds and codimension zero embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The basic building blocks in the construction of the Taylor tower of F are spaces of the form F(� i Rm), for i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='. The homotopy fiber of the map TkF → Tk−1F depends on the total homotopy fiber of the following cubical diagram, indexed by the poset of subsets of k = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}: (1) S �→ F \uf8eb \uf8ed� k\\S Rm \uf8f6 \uf8f8 This homotopy fiber is sometimes called the k-th derivative (or the k-th cross-effect) of F at ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The following fact is particularly important for analysing intrinsic convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Recall that a cubical diagram is called c-cartesian if the map from the initial object to the homotopy limit of the rest of the cubical diagram is c-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose the cubical diagram (1) is ck-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then the map TkF(M) → Tk−1F(M) is ck − mk-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus the Taylor tower of F converges intrinsically at M if the number ck − mk approaches ∞ as k approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When F(M) = Emb(M, Rn), there is a well-known equivalence Emb(� k Rm, Rn) ≃ Conf(k, Rn), where Conf(k, Rn) is the configuration space of ordered k-tuples of pairwise distinct points in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 3 Similarly, there is an equivalence between rImm(� k Rm, Rn) and the so-called r-configuration space, also called no r-equal configuration space, defined by rConf(k, Rn) := rImm(k, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is the space of ordered k-tuples of points in Rn where at most r −1 are allowed to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A proof of the equivalence rImm( � k Rm, Rn) ≃−→ rConf(k, Rn) is given in [AˇS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus r-configuration spaces are basic building blocks in the Taylor tower of rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' To analyse the intrinsic convergence of the Taylor tower of the functor HZ ∧ rImm(−, Rn), one needs to calculate how cartesian the following k-dimensional cubical diagram is (2) S �→ HZ ∧ rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space rConf(i, Rn) is the complement of a subspace arrangement in Rni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the homology of r-configuration spaces is accessible by means of the Goresky-MacPherson formula and other such tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homology of r-configuration spaces was studied by a number of people, starting with Bj¨orner and Welker [BW95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Using the Goresky-MacPherson formula and the results in [BW95] we prove the following result (it is combining Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r ≤ n + 1, the cube (2) is k(n − 1) + � k r � (r − n − 1)-cartesian, and the map pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn) is k � nr − 1 r − m − 1 r � − (k mod r) r (r − n − 1)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Here (k mod r) := k − r � k r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r ≥ n + 1, the cube (2) is k(n − 1) + r − n − 1-cartesian, and the map pk is k(n − m − 1) + r − n − 1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2 follows easily from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 9 we compare the tower of the homological functor HZ ∧ rImm(M, Rn) with that of the tower of the homotopical functor rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us suppose that we chose a basepoint in the space rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this case the presheaf rImm(−, Rn) takes values in pointed spaces, and we have the following diagram of presheaves: (3) rImm(−, Rn) i←− rImm(−, Rn) h−→ Ω∞HZ ∧ rImm(−, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is well-known that the map i induces an equivalence of all layers except the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Indeed, the map i is the homotopy fiber of the map from rImm(−, Rn) to its linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus we can view the map h as a map from the higher layers/derivatives of rImm(−, Rn) to the corresponding layers/derivatives of Ω∞HZ∧rImm(−, Rn), which are essentially the same as the layers/derivatives of HZ ∧ rImm(−, Rn), since Ω∞ commutes with Taylor approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 4 When r = 2, the second derivative of rImm(−, Rn) is equivalent to Sn−1, and the second derivative of HZ ∧ rImm(−, Rn) is HZ ∧ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that in the case r = 2, the map h in (3) induces the Hurewicz homomorphism from the second derivatives of rImm(−, Rn) to the second derivative of HZ ∧ rImm(−, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, it follows that the connectivity of the quadratic layers of the Taylor towers of rImm(−, Rn) and of HZ ∧ rImm(−, Rn) is the same, and their first non-trivial homotopy groups are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By contrast, at degrees higher than 2, the layers of the homotopical tower rImm(−, Rn) and of the homological tower of the functor HZ ∧ rImm(−, Rn) have different connectivities, and there is no Hurewicz type isomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' And again by contrast, in Section 9 we show that for r > 2 the map h in diagram (3) induces a Hurewicz type isomorphism between first non-trivial homotopy groups of layers roughly up to degree 2r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' See Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 for precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 2 we review some background material on cubical diagrams, manifold calculus and spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 3 we introduce the homological Taylor tower that is the main subject of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 4 we make an excursion into the subspace arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We describe r-configuration spaces via subspace arrangements and compute their cohomology using the Goresky-MacPherson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 5 we define the notion of a retractive cubical diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is a diagram where the maps have sections that satisfy a certain hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We prove that the homotopy groups of the total homotopy fiber of a retractive cube are isomorphic to the total kernel of the cube of homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 6 we prove that the cube of r-configuration spaces that controls the layers in the Taylor tower is retractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 7 we prove our main result about the homological connectivity of the cube of r-configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 8 we prove the main result about the intrinsic convergence of the Taylor tower of HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 9 we compare the tower of HZ ∧ rImm(M, Rn) with the tower of rImm(M, Rn) in low degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We prove that the layers in the two towers have the same connectivity up to degree 2r − 1 (with some exceptions in the cases r = 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 10 we discuss some possible directions for further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Prerequisites 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Cubical diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Cubical diagrams play an important role in functor calculus, and in this paper in particular, so we will recall a few elementary facts about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' All the results in this subsection, and much more, can be found in [Goo92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let k denote the standard set with k elements {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let P(k), or just P(k), denote the poset of subsets of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A k-dimensional cubical diagram in a category C is a functor χ: P → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is easy to see that P(k) is equivalent to P(k)op, so a contravariant functor from P(k) to C is called a cubical diagram as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will mostly consider cubical diagrams in (pointed) spaces and spectra, and also in abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 5 Given a cubical diagram χ in topological spaces or spectra, there is a natural map iχ : χ(∅) → holim ∅̸=S⊂k χ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We say that χ is c-cartesian, if this map is c-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homotopy fiber of this map is called the total homotopy fiber of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The total homotopy fiber of χ is denoted by tfiber(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Clearly if χ is c-cartesian then tfiber(χ) is c−1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The converse always holds for cubical diagrams of spectra, and it holds for spaces under the additional assumption that iχ is surjective on path components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One can identify a k-dimensional cubical diagram with a map of two k − 1-dimensional cubical diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Given a k-dimensional cubical diagram χ, let us define two k − 1-dimensional cubical diagrams χ1 and χ2 as follows: χ1(U) = χ(U), and χ2(U) = χ(U ∪ {k}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then χ can be identified with the map of cubes χ1 → χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Furthermore, there is a homotopy fibration sequence whose meaning is that total homotopy fiber can be calculated as an iterated homotopy fiber tfiber(χ) ≃ hofiber(tfiber(χ1) → tfiber(χ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When χ is a cubical diagram of abelian groups, we define the total kernel of χ to be tkernel(χ) := ker(χ(∅) → k � i=1 χ({i})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Just as with total fibers, the total kernel can be calculated as an iterated kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There is a natural isomorphism tkernel(χ) ∼= ker(tkernel(χ1) → tkernel(χ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When χ is a cubical diagram of spaces or spectra, there is a natural homomorphism of graded groups π∗(tfiber χ) → tkernel(π∗χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This homomorphism is not an isomorphism in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In Section 5 we will investigate a condition on a cubical diagram that guarantees for it to be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Manifold calculus of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let M be a smooth manifold of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define O(M) to be the poset category of open subsets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Objects of O(M) are open sets U ⊆ M, and morphisms U → V are the inclusions U ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Manifold calculus of functors, developed by Weiss [Wei99] and Goodwillie-Weiss [GW99], studies contravariant functors from O(M) to a category that supports a reasonable notion of homo- topy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In their foundational papers, Goodwillie and Weiss only considered functors with values in topological spaces, and maybe spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Nowadays it is natural to let the target category to be an ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will content ourselves with functors with values in (pointed) spaces and in spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Technically speaking, manifold calculus applies to functors that are good, in the sense that they satisfy the following two conditions: (i) they are isotopy functors, and (ii) they are finitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A functor is an isotopy functor if it takes isotopy equivalences to weak homotopy equivalences (for the definition of isotopy equivalence see [MV15, Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is finitary if for every 6 monotone union � i Ui (where Ui ⊂ Ui+1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=') the canonical map from F(� i Ui) to holimi F(Ui) is a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If F is a ”half-good” contravariant functor (cofunctor), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' an isotopy functor which is not a finitary functor, then we need to tame this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We call V ∈ O(M) tame if V is the interior of a compact smooth codimension zero submanifold of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' As mentioned in [GKW01], property (ii) ensures that a good cofunctor F on O(M) is essentially determined by its behavior on tame open subsets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, suppose F is a cofunctor from O(M) to Top having property (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then the functor defined by F #(V ) := holimtame U⊂V F(U) for V ∈ O(M) has also property (ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' F # is a good cofunctor on O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We call F # the taming of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There exists a natural transformation F → F #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The map F(V ) → F #(V ) is an equivalence whenever either F or V is tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The motivating example for the development of the manifold calculus of functors is the embedding functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Space of embeddings) Let M and N be smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A smooth embedding of M in N is a smooth map f : M → N such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' the map of tangent spaces Dxf : TxM → Tf(x)N is an injection for all x ∈ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' the derivative of f is a fiberwise injection, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' f : M → f(M) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space of embeddings, Emb(M, N), is the subspace of the space of smooth maps from M to N consisting of smooth embeddings of M in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space Emb(M, N) is topologized using Whitney C∞-topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' for an explanation see [MV15, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' An important example of a space of embeddings with very rich theory is the space of classical knots defined to be the space Emb(S1, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Embedding functor) For a smooth n-dimensional manifold N, the embedding functor Emb(−, N) : O(M) → Top is a contravariant functor given by U �→ Emb(U, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The contravariance follows from the fact that an inclusion of open subsets of a manifold M gives a restriction map of embedding spaces of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A related notion is the space of immersions Imm(M, N), which is a space of smooth maps f : M → N such that just the derivative of f is a fiberwise injection, (property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If M is a compact manifold and f is an injective immersion M → N, then f is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The corresponding functor is the immersion functor Imm(−, N) : O(M) → Top given by U �→ Imm(U, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Functors Emb(−, N) and Imm(−, N) are examples of good functors (see [Wei99] and [GKW01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 7 The idea of the manifold calculus of functors is to approximate a good functor with simpler, polynomial functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Polynomial functor) A good contravariant functor F : O(M) → Top is called polynomial of degree ≤ k if for all U ∈ O(M) and for all pairwise disjoint closed subsets A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', Ak ⊂ U, the (k + 1)-cube P(k + 1) → Top S �→ F(U − � i∈S Ai) is homotopy cartesian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' equivalently, the map F(U) → holimS̸=∅ F(U − � i∈S Ai) is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Here P(k + 1) is the poset category of all subsets of the set k + 1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', k + 1} with ⊂ as the relation of partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Its shape is an (k + 1)-dimensional cubical diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is well known that a polynomial f : R → R of degree k such that f(0) = 0 is uniquely determined by its values on k distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In analogy, a polynomial functor is completely determined by its values on the category of at most k open discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' [Mun10] provides more analogies between the ordinary calculus of functions and the manifold calculus of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More precisely, let Ok(M) be the full subcategory of M consisting of open subsets of M diffeo- morphic to ≤ k disjoint discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We have the following theorem due to Weiss ([Wei99, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose F, G : O(M) −→ Top are good functors that are polynomials of degree ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If T : F → G is a natural transformation that is an equivalence for all U ∈ Ok(M), then T is an equivalence for all U ∈ O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The functor U �→ Imm(U, N) is a polynomial of degree ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The functor U �→ Emb(U, N) is not a polynomial of degree ≤ k for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the details, see [MV15, Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='10], [Wei99, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3], [Mun10, Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Polynomial approximations) For a good functor F, define for each U ∈ O(M) the kth polynomial approximation of F to be TkF(U) = holimV ∈Ok(U) F(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' As Weiss proved in [Wei99, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1], such defined TkF is polynomial of degree ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Also, higher derivatives of such defined polynomial functors vanish and derivatives of a functor and derivatives of its kth polynomial approximation agree up to kth degree, where the derivatives of functors are defined as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Derivative of a functor) Let Dm 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', Dm k be pairwise disjoint open discs in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define a k-cube of spaces by the rule S �→ F(� i/∈S Dm i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We define the kth derivative of F at the empty set, denoted F (k)(∅), to be the total homotopy fiber of the cube S �→ F(� i/∈S Dm i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 8 For example, the 1st derivative of embeddings are immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Also, the linearization of the space of embeddings is the space of immersions, namely there exists an equivalence T1 Emb(−, N) ≃ Imm(−, N) ([Wei99]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For more details and intuition behind this, see Munson’s survey [Mun10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For other relevant results, see [MV15, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='16] and [Wei99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The inclusion Ok−1(U) → Ok(U) induces a map TkF(U) → Tk−1F(U) and so we obtain a tower of functors, called the manifold calculus Taylor tower of F: (4) F(−) �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ � �▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ �❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ T0F(−) · · � Tk−1F(−) � TkF(−) � · · � T∞F(−) � Here T∞F denotes the homotopy inverse limit of this tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' TkF is also called the kth stage of the Tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By evaluating diagram (4) on U ∈ O(M), we get a diagram of spaces with maps between the stages that are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, we can set U = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Layer) Define the kth layer of the manifold calculus Taylor tower of F to be the homotopy fiber of the map between two successive stages of the tower, that is, LkF = hofiber(TkF → Tk−1F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We need to work here with a based Taylor tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It can be accomplished by choosing a basepoint in the space F(M) which then also bases the spaces TkF(U) for all k and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One of the fundamental results, which is a consequence of the Classification of homogeneous functors theorem ([Wei99, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='5], see also [MV15, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='23 and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='26]) is the following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For a good functor F defined on m-dimensional manifolds, if the cube S �→ F \uf8eb \uf8ed� k\\S Dm \uf8f6 \uf8f8 is ck-cartesian, then the map TkF(M) → Tk−1F(M) is ck − km-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More generally, if U has handle dimension j, then the map TkF(U) → Tk−1F(U) is (ck − kj)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the definition of handle dimension, see [MV15, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the Taylor tower of F converges intrinsically at M if the number ck−mk approaches ∞ as k approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The subject of this paper is a functor that represents homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We had a choice between working with chain complexes and the singular chains functor, or working with spectra and using smash product with the Eilenberg-MacLane spectrum to represent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We chose the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 9 We adopt a naive, old-fashioned view of spectra as sequences of spaces equipped with structure maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Spectrum) A spectrum E is a sequence of based spaces {En}n∈N0 together with basepoint-preserving maps (called structure maps) (5) ΣEn → En+1, or, equivalently, the maps (6) En → ΩEn+1, where Σ and Ω denote suspension and loop space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If the maps (6) are weak equivalences, then E is called an Ω-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Each En from an Ω-spectrum is called an infinite loop space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Eilenberg-MacLane spectrum) Let n be an arbitrary positive integer and G be an arbitrary group, abelian for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then there exists a CW complex X such that (7) πn(X) ∼= G and πk(X) is trivial for k ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A topological space X with property (7) is called an Eilenberg-MacLane space K(G, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For example, K(Z, 1) ≃ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For an abelian group G, the Eilenberg-MacLane spectrum, denoted by HG, is defined to be the spectrum {En}n∈N0 with En = K(G, n + 1) and maps (8) K(G, n + 1) → ΩK(G, n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The maps (8) are weak equivalences, hence HG is an Ω-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Since for a spectrum E there exist maps πi+n(En) → πi+n+1(En+1) (for details, see [Hat02, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='F]), it makes sense to define the ith homotopy group of the spectrum E as πi(E) = colimn πi+n(En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A map of spectra f : E → F is a collection of maps fn : En → Fn, n ≥ 0 that commute with the structure maps in E = {En} and F = {Fn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Taking spectra as objects and maps of spectra as morphisms we can define the category of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is denoted by Spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A spectrum can be smashed with a pointed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let E = {En} be a spectrum and X be a based space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The spectrum E ∧ X is defined by (E ∧ X)n = En ∧ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 10 Since Σ(En ∧ X) ∼= (ΣEn) ∧ X, the structure maps in the spectrum E ∧ X are the products of structure maps in E and the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For a spectrum E∧X the homotopy groups πi(E∧X) are the groups colimn πi+n(En∧X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' These groups define a generalized reduced homology theory, determined by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The following result is a consequence of Proposition 4F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2 in [Hat02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' See also [Whi62] for more details on representing generalized homology theories with spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the Eilenberg-MacLane spectrum HZ there exists an isomorphism πi(X ∧ HZ) ∼= �Hi(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If a spectrum E = {En}n≥0 is an Ω-spectrum, then πn(E) is πn(E) = � πn(E0), for n ≥ 0 π0(E−n), for n ≤ 0 Let us note that smash product with a spectrum can be extended from pointed to unpointed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let E be a spectrum and X an unpointed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define the smash product of E and X to be the homotopy fiber of the map E ∧ X+ → E induced by the canonical map X+ → S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For any choice of basepoint in X, there is a canonical equivalence between the new and the old definition E ∧ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' But the new definition does not depend on a choice of basepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is a variant of the fact that reduced homology can be defined as relative homology to a basepoint, but also can be defined independently of basepoint, using the augmented chain complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' However, it is also important to note that without a choice of basepoint in X, there is no natural map X → Ω∞HZ ∧ X representing the Hurewicz homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Such a map is defined only with a choice of basepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We can assume that each spectrum is an Ω-spectrum up to weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Precisely, the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Every spectrum is weakly equivalent to an Ω-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If two spectra E and F are weak equivalent, we write E ≃ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Operation Σ∞ which assigns to a based space X its suspension spectrum Σ∞X, defined by En = ΣnX with identities as structure maps, is a functor Σ∞ : Top∗ → Spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Its adjoint functor Ω∞ : Spectra → Top∗ is defined to be the functor which takes a spectrum E = {En}n≥0, then replaces it by an equivalent Ω-spectrum F = {Fn}n≥0 (which exists using proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='16) and finally picks off the first place F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In short, Ω∞(E) = F0 where F = {Fn}n≥0 ≃ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This F0 is an infinite loop space, which explains the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 11 It follows from the results and comments above that nth homotopy group of a spectrum E equals the nth homotopy group of the space Ω∞(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Finally, let us mention that in addition to the smash product of a spectrum with a space, there is a very important notion of smash product of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For our purposes, the most naive version of the construction suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Given two spectra E = {En} and F = {Fn}, we define their smash product E ∧ F by the formulas (E ∧ F)2n = En ∧ Fn, and (E ∧ F)2n+1 = En+1 ∧ Fn, with the structure maps being induced from the structure maps in E and F in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The sphere spectrum is the unit (up to homotopy) for this smash product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One feature of smash product of spectra that plays a role in this paper is that unlike smash product of spaces, smash product with a fixed spectrum commutes with finite homotopy limits of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More generally, it commutes with homotopy limits over a category whose classifying space is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is discussed in some detail in [LRV03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The significance for us is that if F is a good presheaf of spectra on M, and E is a fixed spectrum, then there are natural equivalences E ∧ TkF ≃ TkE ∧ F and E ∧ LkF ≃ LkE ∧ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homological Taylor tower for reduced r-immersions in Rn The main goal of this paper is to give a convergence result about the homological Taylor tower for the space of r-immersions of a smooth manifold M in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' As is often the case, when studying the homological tower, it is convenient to replace the functor of r-immersions by r-immersions “modulo immersions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This enables us to express the layers in the Taylor tower in terms of r-configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let M be a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Assume that a basepoint in the space Imm(M, Rn) is chosen, and therefore the functor U �→ Imm(U, Rn) is a presheaf of pointed spaces on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Recall that for U ⊂ M, rImm(U, Rn) denotes the homotopy fiber of the map rImm(U, Rn) → Imm(U, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let HZ denote the Eilenberg-Mac Lane spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The functor X �→ HZ ∧ X represents reduced homology, in the sense that there is a natural isomorphism (9) π∗(HZ ∧ X) ∼= �H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Furthermore, recall that the functor can be extended to unpointed spaces, by defining HZ ∧ X for unpointed X to be the homotopy fiber of the map HZ ∧ X+ → HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this paper we study the following functor HZ ∧ rImm(−, Rn): O(M) → Spectra U �→ HZ ∧ rImm(U, Rn) This functor is representing the homology of rImm(−, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 12 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Instead of using spectra and the functor HZ ∧ − to represent homology, we could have used chain complexes and the singular chains functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One reason for choosing spectra is their topological nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The category of spectra, and of HZ-module spectra, is tensored and cotensored over topological spaces, while the category of chain complexes is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Of course, this is a minor technical issue that can be overcome, but anyway it was one reason for us to work with HZ-modules rather than chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Another reason is that working with HZ- modules readily points to generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, most of our results about the functor HZ ∧ rImm(−, Rn) can be extended to the functor Σ∞rImm(−, Rn), which in turn can be used to obtain information about the unstable Taylor tower of rImm(−, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In [GKW01] and [Wei04], Goodwillie, Weiss and Klein point out that for a con- travariant functor F : O(M) → Top, the cofunctor λJF given by U �→ F(U)+ ∧ J for a fixed spectrum J is only ”half-good”, even if F is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Namely, it is an isotopy functor but it fails to be finitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2, to fix this they suggest to use the taming of λJF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will denote the taming of a functor such as λJF by λJF #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The functor λJF # is a good cofunctor, and there is a natural transformation λJF → λJF #, which is an equivalence when evaluated on a tame subset of M, where by a tame subset we mean an open subset which is diffeomorphic to the interior of a compact manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' From now on, whenever we write HZ ∧ rImm(−, Rn) we really mean the taming of this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In practice it makes no difference since we only are interested in evaluating our functors on tame manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So we need to figure out the connectivity of the kth layer of the Taylor tower for the space HZ∧rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='9, this is determined by the homotopy fiber of the cubical diagram, indexed by subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}, S �→ HZ ∧ rImm \uf8eb \uf8ed� k\\S Dm, Rn \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There is a natural map rImm(� k\\S Dm, Rn) → rConf(k \\ S, Rn), which is the composition of the natural map into rImm(� k\\S Dm, Rn), followed by evaluation at the centers of the discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By the main result of [AˇS22], this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the connectivity of the layers of HZ ∧ rImm(M, Rn) is determined by the connectivity of the total fiber of the cubical diagram S �→ HZ ∧ rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' To analyze the total fiber of this cube, we need to review some facts about the homology of r-configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This will be done in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' r-configuration spaces in Rn as complements of subspace arrangements We saw in the previous section that the convergence of the Taylor tower of the functor HZ ∧ rImm(−, Rn) is determined by the homology of r-configuration spaces rConf(k, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' These con- figuration spaces can be interpreted as the complement of an arrangement of subspaces of (Rn)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The combinatorics and topology (in particular, homology and cohomology) of subspace arrange- ments and their complements are well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Some of main references are [OS80], [GM80], [GM83a], [GM83b], [BEZ90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, the (co)homology of r-configuration was studied from 13 this perspective first by Bj¨orner and Welker in [BW95], and by a number of people after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this section we review a qualitative description of the cohomology of r-configuration spaces, based on the Goresky-MacPherson formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will also describe the effect on cohomology of restriction maps between configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Recall that an r-configuration space of k points in Rn is defined to be the space rConf(k, Rn) = {(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', vk) ∈ (Rn)k : ∄1 ≤ i1 < · · · < ir ≤ k such that vi1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' = vir}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space rConf(k, Rn) is an example of the complement of a subspace arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us now recall some formal definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose I is an r-tuple of integers I = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , ir), where 1 ≤ i1 < · · · < ir ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us denote the set of all such r-tuples by �k r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define AI = {(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , vk) ∈ (Rn)k | vi1 = · · · = vir}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let A = � AI | I ∈ �k r �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When we need to make the set k explicit, we write Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More generally, for any set T define AT to be the set of “r-equal” diagonals in (Rn)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Note that one can identify rConf(k, Rn) with the complement of the union of the AIs: rConf(k, Rn) = (Rn)k \\ � I∈(k r) AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If k < r, rConf(k, Rn) ∼= (Rn)k ≃ ∗ If k = r, rConf(k, Rn) ∼= (Rn)r − ∆ ≃ S(r−1)n−1, where ∆ is the thin diagonal in (Rn)r and S(r−1)n−1 is the sphere of dimension (r − 1)n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The collection A of linear subspaces of Rnk is an example of a subspace arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Recall that the intersection lattice of A is the poset LA consisting of all the intersections AI1 ∩· · ·∩AIt of elements of A, ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We include in LA the “empty intersection” of AIs, which is Rnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The space Rnk is the minimal element of LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It will be denoted by ˆ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The maximal element of LA is the intersection of all the AI, which, assuming k ≥ r, is the diagonal copy of Rn in Rnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We denote the maximal elements of LA by ˆ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The poset LA is isomorphic to the poset Πk,r of partitions of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} whose every block is either a singleton or contains at least r elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We call elements of Πk,r r-equal partitions of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The partitions are ordered from finer to coarser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The isomorphism Πk,r → LA sends a partition λ of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} to the space of k-tuples of vectors (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , vk) ∈ (Rn)k with the property that vi = vj whenever i and j are in the same block of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Equivalently, one can say that λ is sent to the space of functions from k to Rn that are constant on each block of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' From now on we will identify the posets LA and Πk,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Because LA is a partially ordered set, we can define the open interval (x, y) in LA to be the set (x, y) = {z ∈ LA | x < z < y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The order complex ∆(x, y) of an open interval (x, y) in LA, is the abstract simplicial complex whose vertices are the elements of (x, y) and whose p-simplices are the chains x0 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' < xp in (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 14 Let �Hi(x, y) denote the ith reduced simplicial homology group of ∆(x, y) with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Similarly, �H i(x, y) denotes the ith reduced cohomology group of ∆(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The (reduced) cohomology groups of the space rConf(k, Rn) = Rnk\\� I∈(k r) AI can be described in terms of (reduced) homology groups of the order complex of intervals in the intersection lattice of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is known as the Goresky-MacPherson formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the original proof of the Goresky- MacPherson formula by means of stratified Morse theory see [GM88, Part III].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' An elementary proof was given by Ziegler and ˇZivaljevi´c in [ZˇZ93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the original calculation of the cohomology rConf(k, Rn) using the Goresky-MacPherson formula see [BW95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Here is the statement, in the case relevant to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4 (Special case of Goresky-MacPherson formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There is an isomorphism (10) �H i(rConf(k, Rn)) ∼= � x∈L>ˆ0 A �Hcodim(x)−2−i(ˆ0, x) Here, the direct sum is indexed by all x ̸= ˆ0 in LA, and codim(x) is the codimension of the space x as the subspace of Rnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For each diagonal x ∈ LA, let c(x) denote the number of components of the partition of k which determines the diagonal x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Obviously, dimension of x in (Rn)k is dim(x) = n · c(x), so (11) codim(x) = n(k − c(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The following easy example of 3-configuration spaces of 4 points illustrates the application of formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let A is the set of all (at least 3)-diagonals in (Rn)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then 3 Conf(4, Rn) = (Rn)4 − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The intersection lattice LA of A is pictured in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4, we find that for every n > 1, H0(3 Conf(4, Rn)) ∼= Z, H2n−1(3 Conf(4, Rn)) ∼= Z4, H3n−2(3 Conf(4, Rn)) ∼= Z3, and other cohomology groups are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For n = 1, the formula is still valid, except that in this case 2n−1 = 3n−2 = 1, so the two cohomology groups add together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So H0(3 Conf(4, R)) ∼= Z and H1(3 Conf(4, R)) ∼= Z7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For n = 1, 2, the cohomology of 3 Conf(4, Rn) can be read off the tables at the end of [BW95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For the purpose of analysing the layers in the homological Taylor tower for r-immersions it also is desirable to know the effect of restriction maps between r-configuration spaces on cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose we have a subset T ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then we have a restriction map rConf(k, Rn) → rConf(T, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We want to describe the induced homomorphism on cohomology, in terms of formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The inclusion T ֒→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} induces an inclusion of the poset of r-equal partitions of T into the poset of r-equal partitions of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}, by making each element of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} \\ T into a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Notice that for every r-equal partition of T, the codimension of the corresponding diagonal is the same whether it is considered a diagonal in (Rn)T or in (Rn)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is so because the codimension of a diagonal determined by a partition is determined by the difference between 15 (1)(2)(3)(4) (1)(2, 3, 4) (1, 2, 3, 4) (2)(1, 2, 3) (3)(1, 2, 4) (4)(1, 2, 3) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Intersection lattice for 3 Conf(4, Rn), also known as Π4,3 the cardinality of the set and the number of blocks of the partition, by formula (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This number remains unchanged if one adds some singletons to a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus we have a homomorphism (12) � x∈L>ˆ0 AT �Hcodim(x)−2−i(ˆ0, x) → � x∈L>ˆ0 Ak �Hcodim(x)−2−i(ˆ0, x) which is defined by the inclusion L>ˆ0 AT ֒→ L>ˆ0 Ak, and uses the fact that for every x ∈ L>ˆ0 AT , the number codim(x) is the same whether x is considered an element of L>ˆ0 AT or of L>ˆ0 Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homomorphism �H i(rConf(T, Rn)) → �H i(rConf(k, Rn)) corresponds, under the isomorphism (10), to the homomorphism (12) that we just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This follows easily from the fact that the Goresky-MacPherson formula is natural with respect to inclusions of subarrangements [Hu94, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1] □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Total fiber of a retractive cubical diagram In general homotopy groups do not commute with total homotopy fibers of cubical diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this section we will show that for a class of cubes that we call retractive they do commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More precisely, we show that for retractive cubes, the homotopy groups of the total fiber are canonically isomorphic to the total kernel of the cube of homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose we have a two-dimensional cubical diagram of spaces or spectra (13) E∅ i∅,1 � i∅,2 � E1 i1,12 � E2 i2,12 � E12 16 Suppose that all the maps in the square (13) have homotopy sections, so that the square of sections E12 s12,1 � s12,2 � E1 s1,∅ � E2 s2,∅ � E0 commutes up to homotopy, and so that the following mixed square E2 i2,12 � s2,∅ � E12 s12,1 � E0 i∅,1 � E1 also commutes up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Note that the vertical maps in the mixed square are sections, while the horizontal maps are from the original square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us call a square (13) with such sections a retractive square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' More generally, let us define a retractive cubical diagram as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let χ be a k-dimensional cubical diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We say that χ is retractive if for every U ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} and every i /∈ U, the map χ(U) → χ(U ∪ {i}) has a homotopy section, the cube of sections commutes up to homotopy, and furthermore whenever U ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k}, and i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} \\ U, with i < j, the following mixed square commutes up to homotopy χ(U ∪ {j}) � � χ(U ∪ {i, j}) � χ(U) � χ(U ∪ {i}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let χ be a retractive k-dimensional cubical diagram of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let E∗ be any homology theory, and let E∗ be a cohomology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then E∗(tfiber χ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' E∗(tfiber χ)) is a direct summand of E∗(χ(∅)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' of E∗(χ(∅))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Moreover, the following natural homomorphism is an isomorphism: E∗(tfiber χ) ∼ =−→ tkernel (E∗χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Similarly, there is a natural isomorphism tcokernel(E∗χ) ∼ =−→ E∗(tfiber χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will prove the claim for homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The proof of the cohomological statement is the same, reversing all arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The proof is by induction on k, starting with with the case k = 1, which is elementary and well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let us review it anyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' A retractive 1-dimensional cube is a map χ(∅) → χ(1), together with a homotopy section χ(1) → χ(∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The total fiber of the cube is the homotopy fiber of the map χ(∅) → χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By homotopy section we mean that the composition χ(1) → χ(∅) → χ(1) is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the composition E∗χ(1) → E∗χ(∅) → E∗χ(1) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' From here it readily follows that the long exact sequence in E∗ associated with the fibration sequence tfiber χ → χ(∅) → χ(1) splits as a 17 direct sum of split short exact sequences in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Furthermore it readily follows that the following homomorphisms are isomorphisms E∗ tfiber χ ∼ =−→ ker (E∗χ(∅) → E∗χ(1)) ∼ =−→ coker (E∗χ(1) → E∗(χ(∅))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Now suppose the lemma holds for cubes of dimension less than k and let χ be a retractive cube of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let χ1 and χ2 be k − 1-dimensional cubes defined by χ1(U) = χ(U) and χ2(U) = χ(U ∪ {k}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then χ can be identified with the natural map of cubes χ1 → χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The cubes χ1 and χ2 are retractive, so by induction hypothesis, the lemma holds for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The retractions do not quite define a map of cubes χ2 → χ1, because we only assumed that the mixed squares commute up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' But they do define a homomorphism of cubes E∗χ2 → E∗χ1, which is a section of the homomorphism of cubes E∗χ1 → E∗χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We have the following diagram E∗ tfiber χ E∗ tfiber χ1 E∗ tfiber χ2 tkernel E∗χ2 tkernel E∗χ1 tkernel E∗χ2 ∼ = ∼ = ∼ = The top row is induced by applying E∗ to a fibration sequence of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The vertical homomor- phisms are isomorphisms by induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the upper right homomorphism is a split surjection, and the top row is a split short exact sequence in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Fur- thermore, E∗ tfiber χ maps isomorphically onto the kernel of the bottom right map, which is tkernel E∗χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The cube of r-configuration spaces is retractive Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The k-cube of spaces S �→ rConf(k \\ S, Rn) is retractive for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let T be a finite set and suppose x /∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Our first step it to construct a section to the restriction map rT∪{x},T : rConf(T ∪ {x}, Rn) → rConf(T, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let p1: Rn → R be projection onto the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Define a map sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' An element of rConf(T, Rn) is a function f : T → Rn with the property that no r points of T go to the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Extend f to a function from T ∪ {x} by sending x to (max{p1f(t) | t ∈ T} + 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In words, x is sent to the point of Rn whose first coordinate is one more than the maximal first coordinate of the existing points, and all other coordinates are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is clear that the image of x is different from all the other points in the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus if f was an r- immersion, then the resulting map T ∪ {x} → Rn is still an r-immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We have defined a 18 map sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is clear that the following composition is the identity (not even just homotopic to the identity but is the actual identity map) rConf(T, Rn) sT,T ∪{x} −−−−−→ rConf(T ∪ {x}, Rn) rT ∪{x},T −−−−−→ rConf(T, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that sT,T∪{x} is a section of rT∪{x},T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Next, we need to show that whenever x, y /∈ T, the following diagram commutes up to homotopy rConf(T, Rn) � � rConf(T ∪ {x}, Rn) � rConf(T ∪ {y}, Rn) � rConf(T ∪ {x, y}, Rn) It is for this step that we need to assume n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let f : T → Rn represent an element of rConf(T, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The images of f in rConf(T ∪ {x, y}, Rn) under the two ways around the diagram are two extensions of f from T to T ∪ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One of the extensions sends x to (max{p1f(t) | t ∈ T} + 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , 0), and sends y to (max{p1f(t) | t ∈ T} + 2, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The other extension does the same thing, with x and y switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is clear that one can write a homotopy between the two maps, by swapping the images of x and y along a circle in the plane spanned by the first two coordinates of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Finally we need to check that the following mixed square commutes up to homotopy rConf(T ∪ {x}, Rn) � � rConf(T, Rn) � rConf(T ∪ {x, y}, Rn) � rConf(T ∪ {y}, Rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This, too, is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In fact, it is easy to check that there is a well-defined straight line homotopy between the two maps around the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We have shown that the section maps that we have defined make the cube of r-configuration spaces and restriction maps between them into a retractive cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Connectivity of the cube of (co)homologies of r-configuration spaces We have seen that the cube of spaces S �→ rConf(k \\ S, Rn), where S ranges over the subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} is retractive (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the cube of spectra obtained by applying the suspension spectrum functor to it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', the cube (14) S �→ Σ∞ rConf(k \\ S, Rn), is also retractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Our goal is to analyse how cartesian is the cube S �→ HZ∧Σ∞ rConf(k\\S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Smash product commutes with total fibers of cubical diagrams of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Therefore, the answer is the same as for the cubical diagram (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' However, we want to use the description of the cohomology of r-configuration spaces given by the Goresky-MacPherson formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The following lemma says that the homology and cohomology groups of the relevant spectrum are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The homology and cohomology groups of the total fiber of (14) are (non-canonically) isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is known, for example by the results of [BW95], that the homology groups of the space rConf(k, Rn), and therefore also of the suspension spectrum of this space, are finitely generated free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Since the cube Σ∞ rConf(k \\ S, Rn) is retractive, it follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2 that the homology of the total fiber of the cube Σ∞ rConf(k \\ S, Rn) is a direct summand of the homology of Σ∞ rConf(k, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Therefore, the homology groups of the total fiber are also finitely generated free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Therefore they are isomorphic to the cohomology groups of the total fiber, by the universal coefficients theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ It follows that the homological connectivity of the total fiber of (14) is equivalent to the coho- mological connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Next, we give a qualitative description of the cohomology of the total fiber, in the style of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let Π≥r(k) denote the set partitions of k with the property that each component has at least r elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', elements of Πk,r without singletons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The i-th cohomology group of the total fiber of the cube (14) is isomorphic to the following direct sum: (15) � x∈Π≥r(k) �Hcodim (x)−2−i(ˆ0, x) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The cube (14) is retractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Using the cohomological part of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2, we conclude that the i-th cohomology of the total fiber is isomorphic to the cokernel of the homomorphism k � i=1 �H i rConf(k \\ {i}, Rn) → �H i rConf(k, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='6, this homomorphism can be identified with the following homomorphism (16) k � i=1 � x∈L>ˆ0 Ak\\{i} �Hcodim(x)−2−i(ˆ0, x) → � x∈L>ˆ0 Ak �Hcodim(x)−2−i(ˆ0, x) The homomorphism maps each summand in the source isomorphically onto a summand in the tar- get (some summands in the source go to the same summand in the target, so the homomorphism is not injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The image of the homomorphism is the sum of terms corresponding to r-equal partitions with at least one singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The cokernel is the direct sum of terms corresponding to r-equal partitions that do not have a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2 that to find how cartesian the cube (14) is, we need to find the smallest i for which the homology group (17) �Hcodim(x)−2−i(ˆ0, x) is non-trivial for some x ∈ Π≥r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Throughout this section, let x be a partition of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} where each block has at least r elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Recall that c(x) denotes the number of blocks of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Note that if k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , kc(x) are the sizes of the blocks of x, then k1 + · · · + kc(x) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let [ˆ0, x] be the closed interval in Πk,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let x be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose x has c(x) blocks, of sizes k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , kc(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then there is an isomorphism of posets [ˆ0, x] ∼= Πk1,r × · · · × Πkc(x),r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The interval [ˆ0, x] consists of r-equal partitions of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' , k} that are refinements of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is the same data as an r-equal partition of each block of x, which is the same as an element of Πk1,r × · · · × Πkc(x),r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ Given a poset P with a minimum and maximum element, let P0 be the poset P with the minimum and maximum removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let x be as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then there is a homotopy equivalence (∗ denotes joint) |∆(ˆ0, x)| ≃ Σc(x)−1|Π0 k1,r| ∗ · · · ∗ |Π0 kc(x),r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This follows from the lemma, and the well-known fact that given two posets P and Q with minimum and maximum objects, there is a homotopy equivalence [Wal88, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 (d)] |(P × Q)0| ≃ Σ|P0| ∗ |Q0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let x be as in the previous lemma and corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then |∆(ˆ0, x)| is homotopy equivalent to a complex of dimension k−c(x)(r−1)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Furthermore, the homology of |∆(ˆ0, x)| in dimension k − c(x)(r − 1) − 2 is non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By the corollary, the space |∆(ˆ0, x)| is homotopy equivalent to Σc(x)−1|Π0 k1,r| ∗ · · · ∗ |Π0 kc(x),r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By the results of [BW95], |Π0 k,r| is homotopy equivalent to a wedge of spheres, not all of the same dimension, and the top homology of this space occurs in dimension k − r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the space Σc(x)−1|Π0 k1,r|∗· · ·∗|Π0 kc(x),r| is a wedge of spheres, with the top homology occurring in dimension c(x) − 1 + (k1 − r − 1) + · · · + (kc(x) − r − 1) + c(x) − 1 = k − c(x)(r − 1) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If r ≤ k < 2r, there is only one summand x in (15) - this is the partition {k}, or in other words the thin diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For this x, dim ∆(ˆ0, x) = k − r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Now we can state and prove the main result of this section Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r ≤ n + 1, the cube (14) is k(n − 1) + � k r � (r − n − 1)-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r ≥ n + 1, the cube (14) is k(n − 1) + r − n − 1-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Note that when r = n+1 both formulas say that the cube (14) is k(n−1)-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Given x, the smallest i for which the homology (17) might be non-trivial is one that satisfies codim(x)−2−i = dim ∆(ˆ0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='5 we have that the smallest i for which the total cokernel (15) might be non-trivial is one that satisfies codim(x) − 2 − i = k − c(x)(r − 1) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Because codim(x) = n(k − c(x)) for x ∈ Π≥r(k), it follows that (18) i = k(n − 1) + c(x)(r − n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 21 We have to see for which x this number i is the smallest possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We distinguish between two overlapping cases, depending on the sign of r − n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 1) When r − n − 1 ≤ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' when r ≤ n + 1, finding i as small as possible is the same as finding x ∈ Π≥r(k) with the biggest number c(x) of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Since all components have to be of the size at least r, the largest number of them is attained when there is a maximum number of them of the size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In that case, c(x) = � k r � , so the smallest i is i = k(n − 1) + �k r � (r − n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So in this case, the cubical diagram (14) is k(n − 1) + � k r � (r − n − 1)-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2) When r − n − 1 ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' r ≥ n + 1, finding i as small as possible is the same as finding x ∈ Π≥r(k) with the smallest number c(x) of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus we need c(x) to be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This x is actually the thin diagonal in the space (Rn)k that corresponds to the partition {k} of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In that case, i = k(n − 1) + r − n − 1, hence (14) is k(n − 1) + r − n − 1-cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Convergence result Let M be a smooth manifold of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Now we finally can calculate the connectivity of the map (19) TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Knowing that ck-connectivity of the total fiber of the cube (14) implies (ck−km+1)-connectivity of the map (19), we can find the conditions under which the Taylor tower converges, using results from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' There are three different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 1) For r − n − 1 < 0, the connectivity of the map (19) is (20) k(n − 1) + �k r � (r − n − 1) − 1 − mk + 1 = k(n − m − 1) + �k r � (r − n − 1) = k(n − m − 1) + �k r − k mod r r � (r − n − 1) = k � n − m − n r − 1 r � − k mod r r (r − n − 1) = k � nr − 1 r − m − 1 r � − k mod r r (r − n − 1) where we noted that � k r � = k/r − (k mod r)/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Note now that −k mod r r (r − n − 1) 22 is nonnegative since r − n − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This means that, as long as nr − 1 r − m − 1 r > 0, the connectivities increase with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2) For r − n − 1 = 0, the connectivity of the map (19) is (21) k(n − 1) − 1 − mk + 1 = k(n − m − 1), which goes to +∞ as k −→ +∞ if n − m − 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 3) For r − n − 1 > 0, the connectivity of the map (19) is (22) k(n − 1) + r − n − 2 − mk + 1 = k(n − m − 1) + r − n − 1, which goes to +∞ as k −→ +∞ if n − m − 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Thus we proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' (Homological convergence of the Taylor tower for r-immersions in Rn) Let M be an m-dimensional smooth manifold and Rn the n-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Assume n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Let rImm(M, Rn) be the space of r-immersions of M in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Consider the map pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' a) For r ≤ n + 1 the map pk is k � nr − 1 r − m − 1 r � − k mod r r (r − n − 1) connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The tower converges intrinsically if n > rm+1 r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' b) For r ≥ n + 1 the map pk is k(n − m − 1) + r − n − 1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The tower converges intrinsically if n > m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Only the assertions regarding intrinsic convergence remain to be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The tower converges intrinsically if the connectivity of pk approaches ∞ with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In the case r ≤ n+1, since (k mod r) is a bounded function of k, this is equivalent to the condition nr−1 r − m − 1 r > 0, which is the same as n > rm+1 r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In the case r ≥ n + 1, the formula for the connectivity of pk clearly tells us that the connectivity goes to ∞ if n > m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Comparing with the unstable tower In this section we will compare the layers, and the connectivities of the maps in the Taylor tower of HZ ∧ rImm(M, Rn) with those in the Taylor tower of the unstabilized functor rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We will show that roughly up to degree 2r − 1 the connectivities of the maps in the two towers are the same, and the first non-trivial homotopy groups of the layers are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 23 In this section, let us assume that we chose a basepoint in rImm(M, Rn) rather than just in Imm(M, Rn), so that the presheaf rImm(−, Rn) takes values in pointed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We have a diagram of presheaves (23) rImm(−, Rn) i←− rImm(−, Rn) s−→ Ω∞Σ∞rImm(−, Rn) h−→ Ω∞HZ ∧ rImm(−, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The map i induces an equivalence of derivatives and layers beyond the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The map h induces the Hurewicz homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, it induces a Hurewicz isomorphism on the first non-trivial homotopy group of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We focus on the question for which k the map s, and therefore also h ◦ s, induces an isomorphism on the first nontrivial homotopy group of the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r = 2, the answer is known to be: only for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We show that for r > 2 the answer is: for all k ≤ 2r − 1, with a small caveat for r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Assume 0 < dim(M) < n, r > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For 1 < k < r, the following maps are equivalences: Tk rImm(M, Rn) → T1 rImm(M, Rn) ≃ Imm(M, Rn) and TkHZ ∧ rImm(M, Rn) → T1HZ ∧ rImm(M, Rn) ≃ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For r ≤ k ≤ 2r − 1, the connectivity of the map Tk rImm(M, Rn) → Tk−1 rImm(M, Rn) is the same as the connectivity of the map TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When r = 3, k = 2r − 1 = 5, the map s, and therefore also h ◦ s in diagram (23), induces an epimorphism on the first non-trivial homotopy group of the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In all other cases when r > 2, r ≤ k ≤ 2r − 1, the maps s and h ◦ s induce an isomorphism on the first non-trivial homotopy group of the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The case k = r, r + 1 of the last assertion of the theorem can be obtained by comparing our Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 with the calculations done in [SˇSV20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The assertion that T1 rImm(M, Rn) ≃ Imm(M, Rn) follows from the fact that when M = Dm, the following maps are equivalences [AˇS22] Emb(Dm, Rn) ≃−→ rImm(Dm, Rn) ≃−→ Imm(Dm, Rn), together with the fact that the functor Imm(−, Rn) is linear, at least on manifolds whose handle dimension is less than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The assertion that both towers are constant for k < r follows from the fact that the derivatives of both functors vanish below degree r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Indeed, the k-th layer in the Taylor tower of rImm(M, Rn) is determined by the following k-dimensional cubical diagram S �→ rImm( � k\\S Dm, Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By the result of [AˇS22], this cubical diagram is equivalent to the diagram S �→ L(Rm, Rn)k\\S × rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 24 Here L(Rm, Rn) is the space of injective linear maps from Rm to Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' This is the “tangential data” of an immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' When k > 1 the tangential data cancels out, and the last cube is as cartesian as the following cube (24) S �→ rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' On the other hand, the k-th layer in the Taylor tower of Ω∞Σ∞ rImm(M, Rn) is determined by the following k-dimensional cubical diagram (25) S �→ Ω∞Σ∞ rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' To prove the assertion about the connectivities of the maps in the two towers, we need to show that the cubical diagram (24) is as cartesian as the diagram (25) in the indicated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Furthermore, we want to prove that the map s in (23) induces an isomorphism/epimorphism on the first non-trivial homotopy groups of the total homotopy fibers in the appropriate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The map s induces the following map of cubical diagrams, indexed by the poset of subsets S ⊂ k, (26) rConf(k \\ S, Rn) s−→ Ω∞Σ∞ rConf(k \\ S, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The spaces rConf(k \\ S, Rn) are (r − 1)n − 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' By Freudenthal suspension theorem, the maps (26) are 2(r−1)n−3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' On the other hand, both cubes are retractive cubes by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It follows that the homotopy groups of the total homotopy fibers of both cubes are isomorphic to the total kernels of the corresponding cubes of homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 tells us the connectivity of the total homotopy fiber of (25), and therefore also the connectivity of the total kernel of the corresponding cube of homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' If this connectivity is smaller than (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' equals to) the connectivity of the maps in (26), then (26) induces an isomorphism (resp: an epimorphism) between the first non-trivial homotopy groups of the total homotopy fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So we have to check that the range provided by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 is smaller than (or equals to) 2(r − 1)n − 3 in the cases indicated in the statement that we are trying to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Suppose first that r > n+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this case, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 says that (25) is k(n−1) + r −n−1- cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So we have to check that the inequality k(n − 1) + r − n − 1 < 2(r − 1)n − 3 holds whenever k < 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Simplifying, we obtain the inequality k < (2n − 1)r − 3 n − 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So it is enough to check the inequality 2r ≤ (2n − 1)r − 3 n − 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Multiplying by n − 1 we obtain the inequality 2r(n − 1) ≤ (2n − 1)r − 3 − n + 1, which is equivalent to r ≥ n + 2, which is what we assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Now suppose that r ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='7 says that (25) is k(n − 1) + � k r � (r − n − 1)- cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So we have to check that the inequality k(n − 1) + �k r � (r − n − 1) < 2(r − 1)n − 3 25 holds when r ≤ k < 2r, with the exception that when r = 3, k = 5 it is in fact an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The reader can check that in this case we do indeed obtain the equality 5(n − 1) + �5 3 � (3 − n − 1) = 4n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In other cases, the assumption r ≤ k < 2r implies � k r � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' So we have to check the inequality k(n − 1) + r − n − 1 < 2(r − 1)n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We can rewrite the inequality as follows k(n − 1) < (2r − 1)(n − 1) + r − 3, or equivalently k < 2r − 1 + r − 3 n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For r = 3 this inequality is equivalent to k < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For 3 < r ≤ n + 1, this holds for all k ≤ 2r − 1, as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Further questions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We gave conditions on the m and n that guarantee intrinsic convergence of the Taylor tower of HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The next question is, what does the Taylor tower from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 converge to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is natural to guess that whenever the Taylor tower converges intrinsically, it actually converges to HZ ∧ rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' What can one say about the convergence of the Taylor tower for the unstable functor rImm(M, Rn)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The question of intrinsic convergence of the unstable tower might be tractable, and is a good place to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' One can use the methods of this paper to describe the layers of the functor Σ∞rImm(M, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Given this, one can try to analyse the layers of the functor rImm(M, Rn) via the cobar construction cobar(Ω∞, Σ∞Ω∞, Σ∞rImm(M, Rn)), in the style of [AC11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' It is conceivable that one can use these methods to obtain conditions on m, n, and r that guarantee that the tower converges intrinsically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Then there is a question of what the tower actually converges to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Once again, it seems reasonable to guess that whenever the Taylor tower of a “natural” functor converges intrinsically, then it actually converges to the functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' What can one say about r-immersions into a general manifold N?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In order to understand the layers of the tower of the functor Σ∞rImm(M, N) one needs to understand (the stable homotopy type of) the homotopy fiber of the map rConf(k, N) → Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' For r = 2 this homotopy fiber was analysed in [Aro09], and it seems likely that a similar analysis can be done for general r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Construct interesting invariants/obstructions to existence of r-immersions, using the Taylor tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In this paper we focused on situations where the connectivity of the k-th layer in the tower goes to infinity as k goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' But situations when the connectivity does not go to infinity also can be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Of particular potential interest are situations where the layers are all either −1-connected or −2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In the former case, the bottom homotopy groups of the layers give invariants, in the latter case they give obstructions to existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' 26 For example, it follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 that when n = m+1 and r = n+1, then all the layers of HZ∧(n+ 1) Imm(M, Rn) are −1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' The 0-th homotopy groups of the layers should give invariants of r-immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In the case n = 2, and say M = S1, 3 Imm(S1, R2) is the space of smooth curves in R2 that do not have triple intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Spaces of such curves were studied quite intensely, starting with Arnol’d [Arn94, Tab96, Shu95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular, Arnol’d developed the theory of finite type invariants for such curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' We expect these invariants to show up in the Taylor tower of HZ ∧ 3 Imm(S1, R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' In particular we speculate that the first non-trivial layer of the tower, which by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='1 is the third layer, detects the “Strangeness” invariant, defined in [Arn94] and studied further in [Tab96] and [Shu95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' References [AC11] Greg Arone and Michael Ching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Operads and chain rules for the calculus of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Ast´erisque 338:vi+158, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' [ALV07] Gregory Arone, Pascal Lambrechts, and Ismar 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55(4):499–504, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' [Whi62] George W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Whitehead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Generalized Homology Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', 102(2):227–283, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' [ZˇZ93] G¨unter Ziegler and Rade ˇZivaljevi´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Homotopy types of subspace arrangements via diagrams of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=', 295:527–548, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content=' Gregory Arone Department of Mathematics, Stockholm University Email address: gregory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='arone@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='se Franjo ˇSarˇcevi´c Department of Mathematics, University of Sarajevo Email address: franjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='sarcevic@live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='de URL: pmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='unsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} +page_content='ba/franjos 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfXwS1/content/2301.03131v1.pdf'} diff --git a/INE3T4oBgHgl3EQfugsS/vector_store/index.faiss b/INE3T4oBgHgl3EQfugsS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..df37e6cb20377b082790c86a1b85f9b5e100b997 --- /dev/null +++ b/INE3T4oBgHgl3EQfugsS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aaaeef02dc8465720e976678e21ac4b2611a2352463c6d4fdd2c18056ed5c6b9 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We show the insolvability of the Diophantine equation axd −y2 −z2 +xyz − +b = 0 in Z for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, where d is an +odd integer and is a multiple of 3. Further, we investigate the more general family with +b = 2da − 3r, where r is a positive odd integer. As a consequence, we found an infinite +family of hyperelliptic curves with trivial torsion over Q. We conclude by providing some +numerical evidence corroborating the main results. +1. Introduction +One of the earliest topics in number theory is the study of Diophantine equations. In +the third century, Greek mathematician Diophantus of ‘Alexandria’ began this study. A +polynomial equation of the form +P(x1, x2, · · · , xn) = 0 +is known as a Diophantine equation. Finding all of its integer solutions, or all of the +n−tuples (x1, x2, · · · , xn) ∈ Z that satisfy the above equation, is of prime interest. The +main task is to investigate whether solutions exist for a given Diophantine equation. If +they do, it would be the aim to know how many are there and how to find all. There +are certain Diophantine equations which has no non zero integer solutions, for example, +Fermat’s equation xn + yn = zn for n ≥ 3. The tenth of Hilbert’s 23 problems, which +he presented in 1900, dealt with Diophantine equations. Hilbert asked, is there an al- +gorithm to determine weather a given Diophantine equation has a solution or not? and +Matiyasevich in 1970 answered it negatively. +We investigate a class of Diophantine equations of the form axd−y2−z2+xyz−b = 0 for +fixed a and b. Due to its emergence when attempting to solve an equation involving fruits, +this type of Diophantine equations were given the name “Fruit Diophantine equation” by +B. Sury and D. Majumdar [5] and they proved the following: +2010 Mathematics Subject Classification. Primary: 11D41, 11D72. Secondary: 11G30. +Key words and phrases. Diophantine equation, Quadratic residue, Elliptic curves, Hyperelliptic curves. +1 + +2 +OM PRAKASH AND KALYAN CHAKRABORTY +Theorem 1.1. [5] The equation +y2 − xyz + z2 = x3 − 5 +has no integer solution in x, y and z. +Similar type of equations were previously studied by F. Luca and A. Togb´e. In particu- +lar, Luca and Togb´e [4] studied the solution of the Diophantine equation x3+by+1−xyz = +0 and later, Togb´e [7] independently studied the equation x3 + by + 4 − xyz = 0. +As a consequence of Theorem 1.1 Majumdar and Sury proved the following: +Theorem 1.2. [5] For any integer m, the elliptic curve +Em : y2 − mxy = x3 + m2 + 5 +has no integral point. +L. Vaishya and R. Sharma expanded on Majumdar and Sury’s work in [8]. A class of +fruit Diophantine equations without an integer solution was found by them. In particular +Vaishya and Sharma showed, +Theorem 1.3. [8] For fixed integers a and b with a ≡ 1 (mod 12) and b = 8a − 3. The +Diophantine equation +ax3 − y2 − z2 + xyz − b = 0 +has no integer solution. +Using Nagell-Lutz theorem [6] and Theorem 1.3 they got hold of an infinite family of +elliptic curves with torsion-free Mordell-Weil group over Q. +Theorem 1.4. [8] Let a and b be as in Theorem 1.3. +• For any even integer m the elliptic curve +Ee +m,a,b : y2 = x3 + 1 +4m2x2 − a2 � +m2 + b +� +has torsion-free Mordell-Weil group. +• For any odd integer m the elliptic curve +Eo +m,a,b : y2 = x3 + m2x2 − 64a2 � +m2 + b +� +has torsion-free Mordell-Weil group. + +GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES +3 +We extend Vaishya and Sharma’s results [8] for higher exponents. We obtain a family +of hyperelliptic curves, by carrying out some appropriate transformations. In 2013, D. +Grant gave an analogue of Nagell-Lutz theorem for hyperelliptic curves [3], using which +we conclude that the Mordell-Weil group of each member of the corresponding family of +hyperelliptic curves is torsion-free. +2. Insolvability +Here we state and prove the main theorem and derive a couple of interesting corollaries. +We end this section by looking into a couple of examples. +Theorem 2.1. The equation +axd − y2 − z2 + xyz − b = 0 +has no integer solutions for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, +where d is an odd integer and divisible by 3. +Proof. Consider +axd − y2 − z2 + xyz − b = 0. +(2.1) +If possible, let (x, y, z) be an integer solution of (2.1). Let us fix x = α. Then (2.1) can +be re-written as, +y2 + z2 + b = aαd + αyz. +(2.2) +We consider the cases of α being even or odd separately. +Case 1. If α is even. Then, we write (2.2) as: +� +y − αz +2 +�2 +− +�α2 +4 − 1 +� +z2 = aαd − b +(2.3) +and set Y = y − αz +2 , β = α +2 and z = Z. Thus (2.3) becomes, +Y 2 − +� +β2 − 1 +� +Z2 = aαd − b = 2dβda − b. +(2.4) +• If β is even, say β = 2n for some integer n, then reducing (2.4) modulo 4 gives, +Y 2 + Z2 ≡ 3 +(mod 4), +(2.5) +which is not possible in Z/4Z. + +4 +OM PRAKASH AND KALYAN CHAKRABORTY +• If β is odd, then β = 2n + 1 for some integer n. Reduction of (2.4) modulo 4 +entails, +Y 2 ≡ 3 +(mod 4) +(2.6) +which is impossible. +Case 2. If α is odd, say, α = 2n + 1 for some integer n. Then, +y2 + z2 + b += +aαd + αyz +y2 + z2 + a2d − 3 += +a (2n + 1)d + αyz +y2 + z2 − (2n + 1) yz += +a (2n + 1)d − a2d + 3. +Now +y2 + z2 + yz +≡ +a + 3 +(mod 2), +⇒ y2 + z2 + yz +≡ +0 +(mod 2). +Note that y2 + z2 + yz ≡ a + 3 (mod 2) has only solution y ≡ 0 ≡ z in Z/2Z, that is, y +and z are even. Thus (2.3) becomes +aαd − b ≡ 0 +(mod 4). +If we write a = 12l + 1 for some integer l, then, +αd − +� +a2d − 3 +� +≡ +0 +(mod 4), +⇒ αd + 3 +≡ +0 +(mod 4), +⇒ αd +≡ +1 +(mod 4), +⇒ α +≡ +1 +(mod 4). +Let us consider +� +y − αz +2 +�2 +− +�α2 +4 − 1 +� +z2 += +aαd − b, +i.e. +� +y − αz +2 +�2 +− +� +α2 − 4 +� �z +2 +�2 += +aαd − b. +Further, we set Y = y − αz +2 and Z = z +2. Then, +Y 2 − +� +α2 − 4 +� +Z2 = aαd − b +(2.7) +where α ≡ 1 (mod 4), a ≡ 1 (mod 12) and b = a2d − 3. Three sub cases need to be +considered. + +GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES +5 +Sub-case 1. If α ≡ 1 (mod 12), write α = 12l + 1 for some integer l. Then, +α ≡ 1 +(mod 3) +⇒ α + 2 ≡ 0 +(mod 3). +Substituting α = 12l + 1 in 2.7, we get +Y 2 − +� +(12l + 1)2 − 4 +� +Z2 += +aαd − b, +⇒ Y 2 ≡ aαd − b +(mod 3), +⇒ Y 2 ≡ a (12l + 1)d − a2d + 3 +(mod 3), +⇒ Y ≡ 1 − 2d +(mod 3), +⇒ Y 2 ≡ 2 +(mod 3). +A contradiction as 2 is not square modulo 3. +Sub-case 2. If α ≡ 9 (mod 12). Then, there is a prime factor p ≡ 5 or 7 (mod 12) of +(α − 2). Let p ≡ 5 or 7 (mod 12) be a prime factor of (α − 2). Thus, +Y 2 ≡ aαd − b +(mod p). +Let α = pl + 2 for some integer l. Then, +Y 2 +≡ +a (pl + 2)d − b +(mod p), +⇒ Y 2 +≡ +3 +(mod p). +This leads to a contradiction as 3 is not a quadratic residue modulo p. +Sub-case 3. When α ≡ 5 (mod 12), we substitute α = 3k + 2 for some integer k and +get, +Y 2 − +� +(3l + 2)2 − 4 +� +Z2 += +(12l + 1) (3k + 2) − 2d (12l + 1) + 3, +⇒ Y 2 +≡ +2 − 2d ≡ 0 +(mod 3), +⇒ Y +≡ +0 +(mod 3). + +6 +OM PRAKASH AND KALYAN CHAKRABORTY +Further, we substitute Y = 3m and α = 12n + 5 for some integers n and m in 2.7 and +arrive onto, +9m2 − (12n + 3) (12n + 7) Z2 += +a (12n + 5)d − b, +⇒ − (n + 1) Z2 +≡ +d−1 +� +i=0 +(12n + 5)d−1−i 2i +(mod 3), +⇒ − (n + 1) Z2 +≡ +1 +(mod 3), +⇒ n +≡ +1 +(mod 3). +Hence, α ≡ 17 (mod 36). +Note that 3 divides (α − 2). Thus there is a prime factor p ≡ 5 or 7 (mod 12) of (α−2) +3 +, +otherwise it would mean that α−2 +3 +is congruent to ±1, which is not the case. Therefore, +α − 2 ≡ 0 +(mod p). +Thus, +Y 2 ≡ aαd − b +(mod p). +Substituting α = pl + 2 for some integer l, we have +Y 2 ≡ 3 +(mod p), +which contradicts the fact that 3 is quadratic residue modulo p if p ≡ ±1 (mod 12). +□ +Remark 1. The result of Sury and Majumdar [5] follows by substituting a = 1 and d = 3 +in Theorem 2.1. The particular case d = 3 in the same theorem deduces the results of +Vaishya and Sharma [8]. +By increasing the exponents in the expression for b to 3, we will now examine the +Diophantine equation with a little more generality. The potential of a solution in this +scenario is described by the following two corollaries, along with a few examples. +Corollary 2.1. The equation +axd − y2 − z2 + xyz − b = 0 +has no integer solution (x, y, z) with x even for fixed integers a and b such that a ≡ 1 +(mod 12) and b = 2da − 3r with positive odd integers r and d as in Theorem 2.1. + +GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES +7 +Proof. We follow exactly the same steps as in Case 1 of Theorem 2.1. Suppose there is a +solution with x = α even, then we write (2.2) as: +� +y − αz +2 +�2 +− +�α2 +4 − 1 +� +z2 = aαd − b. +(2.8) +Let Y = y − αz +2 , β = α +2 and z = Z. Then (2.8) can be written as, +Y 2 − +� +β2 − 1 +� +Z2 = aαd − b = 2dβda − b. +(2.9) +• If β is even, say β = 2n for some integer n, then the reduction modulo 4 of (2.9) +will give, +Y 2 + Z2 ≡ 3r ≡ 3 +(mod 4), +(2.10) +which is not feasible in Z/4Z. +• If β is odd, say β = 2n + 1 for some integer n. Then, the reduction modulo 4 of +(2.9) provides, +Y 2 ≡ 3r ≡ 3 +(mod 4), +(2.11) +which again is not possible. +□ +The following corollary deals with solutions having x, an odd integer: +Corollary 2.2. The equation +axd − y2 − z2 + xyz − b = 0 +has no integer solution in x, y and z with x ≡ 1 or 9 (mod 12), for fixed integers a, b +such that a ≡ 1 (mod 12) and b = 2da − 3r, for r and d as in Corollary 2.1. +Proof. Analogous steps as in Sub-case 2 and 3 of Theorem 2.1 will give the proof. +□ +Remark 2. Corollary 2.2 says that, if there is a solution of axd − y2 − z2 + xyz − b = 0 +with a and b as described in the Corollary 2.2, then x must be 5 modulo 12. +We will see some examples. +Example 1. For a = 25, d = 3 and r = 3. The equation +25x3 − y2 − z2 + xyz − 173 = 0 +(2.12) +has no integer solution. + +8 +OM PRAKASH AND KALYAN CHAKRABORTY +Example 2 shows that the equation may not have solution even with x ≡ 5 (mod 12). +However, the next examples tell us the other possibility as well. +Example 2. If a = 13, d = 3 and r = 3, then +13x3 − y2 − z2 + xyz − 77 = 0 +(2.13) +has an integer solution (5, = 18, −102). +Remark 3. The condition that r should be odd is rigid. +Example 3. For a = 13, d = 3 and r = 2, the equation +13x3 − y2 − z2 + xyz − 95 = 0 +(2.14) +has an integer solution (2, −10, −7). +3. Hyperelliptic curves +A hyperelliptic curve H over Q is a smooth projective curve associated to an affine plane +curve given by the equation y2 = f (x), where f is a square-free polynomial of degree at +least 5. If the degree of f is 2g + 1 or 2g + 2, then the curve has genus g. We write H (Q) +for the set of Q-points on H. Determining rational points on hyperelliptic curve is one +of the major problems in mathematics. The following is the general result regarding the +size of H (Q), which was conjectured by Mordell and was proved by Faltings: +Theorem 3.1. [2] If C is a smooth, projective and absolutely irreducible curve over Q of +genus at least 2, then C (Q) is finite. +We may thus, at least theoretically, write down the finite set C (Q). It is still a signifi- +cant unresolved problem to perform this practically for a given curve. +Given a hyperelliptic curve H, we can define the height (classical) function to be the +maximum of absolute values of the coefficients. The Northcott property tells us that there +are finitely many equations with bounded height. Thus, one may talk about the density +and averages. In this regard, Bhargava [1] has proved that most of the hyperelliptic curve +over Q has no rational point. So, most of the times calculating H (Q) means proving +H (Q) = φ. +In this section, we construct hyperelliptic curves corresponding to the equation axd − +y2 − z2 + xyz − b = 0 with a and b as mentioned in Theorem 2.1. +Then, we prove +that H (Q) = φ (corroborating Bhargava [1]). The main ingredient to prove this is the +following Nagell-Lutz type theorem (Theorem 3, [3]) proved by D. Grant. + +GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES +9 +Theorem 3.2. [3] Let C be a nonsingular projective curve of genus g ≥ 1 given by +y2 = x2g+1 + b1x2g + · · · + b2gx + b2g+1, where bi ∈ Z. Suppose +ψ : C (Q) → J (Q) +be the Abel-Jacobi map, defined by ψ (p) = [p − ∞], where J (Q) is the Jacobian variety. +If p = (x, y) ∈ C (Q) \ {∞} and ψ (p) ∈ J (Q)tors, then, x, y ∈ Z and either y = 0 or y2 +divides discriminant of the polynomial x2g+1 + b1x2g + · · · + b2gx + b2g+1. +For fixed m we define hyperelliptic curves, +Hm,a,b : y2 − mxy = axd − m2 − b. +• Suppose m is even. Then write (2.1) as: +� +y − mx +2 +�2 +− m2x2 +4 += axd − m2 − b. +(3.1) +Multiplying (3.1) by ad−1 throughout, and using the fact that d is odd and divisible +by 3, we have, +�� +y − mx +2 +� +a +d−1 +2 +�2 +− ad−1m2x2 +4 += (ax)d − m2ad−1 − bad−1. +(3.2) +We get the following hyperelliptic curve by substituting +�� +y − mx +2 +� +a +d−1 +2 +� += Y and +ax = X, +He +m,a,b : Y 2 − ad−3m2X2 +4 += Xd − m2ad−1 − bad−1. +(3.3) +• Now if m is odd, multiply (3.2) by 4d throughout to get +�� +y − mx +2 +� +a +d−1 +2 2d�2 +− (4a)d−1 m2x2 = (4ax)d − m2ad−14d − bad−14d. +Finally substitute +�� +y − mx +2 +� +a +d−1 +2 2d� += Y and 4ax = X, to get +Ho +m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d. +(3.4) +Let, +Hm,a,b = + + + +He +m,a,b +if m is even +Ho +m,a,b +if m is odd, +(3.5) +be the hyperelliptic curves. +Theorem 3.3. Let a and b be as defined in Theorem 2.1. For any m ∈ N, the hyperelliptic +curve Hm,a,b has torsion-free Mordell-Weil group over Q. + +10 +OM PRAKASH AND KALYAN CHAKRABORTY +Proof. Let a and b be fixed positive integers with a ≡ 1 (mod 12) and b = 2da − 3. +• For any even integer m, consider the hyperelliptic curve +He +m,a,b : Y 2 − ad−3m2X2 +4 += Xd − m2ad−1 − bad−1. +(3.6) +By Theorem 3 of [3], if (3.6) has an integer solution (X0, Y0), +then +� +aX0, +�� +Y0 − mX0 +2 +� +a +d−1 +2 +� +, m +� +is a solution of (2.1). However, in Theorem +2.1 we have proved that it has no integer solutions. +• For an odd integer m, consider the hyperelliptic curve +Ho +m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d. +(3.7) +Suppose (3.7) has a solution (X0, Y0), then +� +4aX0, +�� +Y0 − mX0 +2 +� +a +d−1 +2 2d� +, m +� +is a +solution of (2.1), which is a contradiction. +□ +4. Numerical examples +In this section we give some numerical examples corroborating our results in Corollary +2.2 and Remark 2. +a +d +r +Equation +Solution +1 +3 +3 +x3 − y2 − z2 + xyz + 19 = 0 +(5, 0, −12) +1 +3 +5 +x3 − y2 − z2 + xyz + 235 = 0 +(29, 12, −60) +1 +3 +7 +x3 − y2 − z2 + xyz + 2179 = 0 +(5, 0, −48) +1 +3 +9 +x3 − y2 − z2 + xyz + 19675 = 0 +(−31, 12, −30) +13 +3 +3 +13x3 − y2 − z2 + xyz − 77 = 0 +(5, −18, −102) +13 +3 +5 +13x3 − y2 − z2 + xyz + 139 = 0 +(5, 0, −42) +13 +3 +7 +13x3 − y2 − z2 + xyz + 2083 = 0 +? +25 +3 +3 +25x3 − y2 − z2 + xyz − 173 = 0 +(5, 0, −42) +Acknowledgement +This work is done during the first author’s visit to Institute of Mathematical Sci- +ences (IMSc), Chennai, and he is grateful to the Institute for the hospitality and the +wonderful working ambience. Both the authors are grateful to Kerala School of Mathe- +matics(KSoM), Kozhikode, for it’s support and wonderful ambience. + +GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES +11 +References +[1] M. Bhargava, Most hyperelliptic curve over Q have no rational point, arXiv:1308.0395. +[2] G. Faltings, “Finiteness theorems for abelian varieties over number fields”, Invent. Math., 73 (1983), +349–366. +[3] D. Grant, On an analogue of the Lutz-Nagell theorem for hyperelliptic curves, J. Number Theory, +133 (2013), 963–969. +[4] F. Luca and A. Togb´e, On the positive integral solution of the Diophantine equation x3+by+1−xyz, +Bull. Malays. Math. Sci. Soc., 31 (2008), 129–134. +[5] D. Majumdar and B. Sury, Fruit Diophantine Equation,https://arxiv.org/abs/2108.02640. +[6] J.H. Silverman, , J.T. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics. +Springer-Verlag, New York (1992). +[7] A. Togb´e, On the positive integral solution of the Diophantine equation x3 + by + 4 − xyz, Afr. +Diaspora J. Math., 8 (2009), 81–89. +[8] L. Vaishya and R. Sharma, A class of fruit Diophantine equations, Monatshefte f¨ur Mathematik, 199 +(2022), 899–907. +Kerala School of Mathematics, Kozhikode - 673571, Kerala, India. +Email address: omprakash@ksom.res.in +Kerala School of Mathematics, Kozhikode - 673571, Kerala, India. +Email address: kalychak@ksom.res.in + diff --git a/KtFRT4oBgHgl3EQfDzfl/content/tmp_files/load_file.txt b/KtFRT4oBgHgl3EQfDzfl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55b7f7eff32cc048815fad024d9ba8f6c6789f41 --- /dev/null +++ b/KtFRT4oBgHgl3EQfDzfl/content/tmp_files/load_file.txt @@ -0,0 +1,291 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf,len=290 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='13474v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='NT] 31 Jan 2023 GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES OM PRAKASH AND KALYAN CHAKRABORTY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We show the insolvability of the Diophantine equation axd −y2 −z2 +xyz − b = 0 in Z for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, where d is an odd integer and is a multiple of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Further, we investigate the more general family with b = 2da − 3r, where r is a positive odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We conclude by providing some numerical evidence corroborating the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Introduction One of the earliest topics in number theory is the study of Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In the third century, Greek mathematician Diophantus of ‘Alexandria’ began this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' A polynomial equation of the form P(x1, x2, · · · , xn) = 0 is known as a Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Finding all of its integer solutions, or all of the n−tuples (x1, x2, · · · , xn) ∈ Z that satisfy the above equation, is of prime interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The main task is to investigate whether solutions exist for a given Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If they do, it would be the aim to know how many are there and how to find all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' There are certain Diophantine equations which has no non zero integer solutions, for example, Fermat’s equation xn + yn = zn for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The tenth of Hilbert’s 23 problems, which he presented in 1900, dealt with Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Hilbert asked, is there an al- gorithm to determine weather a given Diophantine equation has a solution or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' and Matiyasevich in 1970 answered it negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We investigate a class of Diophantine equations of the form axd−y2−z2+xyz−b = 0 for fixed a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Due to its emergence when attempting to solve an equation involving fruits, this type of Diophantine equations were given the name “Fruit Diophantine equation” by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Sury and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Majumdar [5] and they proved the following: 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Primary: 11D41, 11D72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Secondary: 11G30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Diophantine equation, Quadratic residue, Elliptic curves, Hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 1 2 OM PRAKASH AND KALYAN CHAKRABORTY Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [5] The equation y2 − xyz + z2 = x3 − 5 has no integer solution in x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Similar type of equations were previously studied by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Luca and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Togb´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In particu- lar, Luca and Togb´e [4] studied the solution of the Diophantine equation x3+by+1−xyz = 0 and later, Togb´e [7] independently studied the equation x3 + by + 4 − xyz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' As a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1 Majumdar and Sury proved the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [5] For any integer m, the elliptic curve Em : y2 − mxy = x3 + m2 + 5 has no integral point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Vaishya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Sharma expanded on Majumdar and Sury’s work in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' A class of fruit Diophantine equations without an integer solution was found by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In particular Vaishya and Sharma showed, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [8] For fixed integers a and b with a ≡ 1 (mod 12) and b = 8a − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The Diophantine equation ax3 − y2 − z2 + xyz − b = 0 has no integer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Using Nagell-Lutz theorem [6] and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3 they got hold of an infinite family of elliptic curves with torsion-free Mordell-Weil group over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [8] Let a and b be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For any even integer m the elliptic curve Ee m,a,b : y2 = x3 + 1 4m2x2 − a2 � m2 + b � has torsion-free Mordell-Weil group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For any odd integer m the elliptic curve Eo m,a,b : y2 = x3 + m2x2 − 64a2 � m2 + b � has torsion-free Mordell-Weil group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES 3 We extend Vaishya and Sharma’s results [8] for higher exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We obtain a family of hyperelliptic curves, by carrying out some appropriate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In 2013, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Grant gave an analogue of Nagell-Lutz theorem for hyperelliptic curves [3], using which we conclude that the Mordell-Weil group of each member of the corresponding family of hyperelliptic curves is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Insolvability Here we state and prove the main theorem and derive a couple of interesting corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We end this section by looking into a couple of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The equation axd − y2 − z2 + xyz − b = 0 has no integer solutions for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, where d is an odd integer and divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Consider axd − y2 − z2 + xyz − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1) If possible, let (x, y, z) be an integer solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let us fix x = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1) can be re-written as, y2 + z2 + b = aαd + αyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2) We consider the cases of α being even or odd separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If α is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, we write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2) as: � y − αz 2 �2 − �α2 4 − 1 � z2 = aαd − b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3) and set Y = y − αz 2 , β = α 2 and z = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3) becomes, Y 2 − � β2 − 1 � Z2 = aαd − b = 2dβda − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='4) If β is even, say β = 2n for some integer n, then reducing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='4) modulo 4 gives, Y 2 + Z2 ≡ 3 (mod 4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='5) which is not possible in Z/4Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 4 OM PRAKASH AND KALYAN CHAKRABORTY If β is odd, then β = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Reduction of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='4) modulo 4 entails, Y 2 ≡ 3 (mod 4) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='6) which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If α is odd, say, α = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, y2 + z2 + b = aαd + αyz y2 + z2 + a2d − 3 = a (2n + 1)d + αyz y2 + z2 − (2n + 1) yz = a (2n + 1)d − a2d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Now y2 + z2 + yz ≡ a + 3 (mod 2), ⇒ y2 + z2 + yz ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Note that y2 + z2 + yz ≡ a + 3 (mod 2) has only solution y ≡ 0 ≡ z in Z/2Z, that is, y and z are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3) becomes aαd − b ≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If we write a = 12l + 1 for some integer l, then, αd − � a2d − 3 � ≡ 0 (mod 4), ⇒ αd + 3 ≡ 0 (mod 4), ⇒ αd ≡ 1 (mod 4), ⇒ α ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let us consider � y − αz 2 �2 − �α2 4 − 1 � z2 = aαd − b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' � y − αz 2 �2 − � α2 − 4 � �z 2 �2 = aαd − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Further, we set Y = y − αz 2 and Z = z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, Y 2 − � α2 − 4 � Z2 = aαd − b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='7) where α ≡ 1 (mod 4), a ≡ 1 (mod 12) and b = a2d − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Three sub cases need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES 5 Sub-case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If α ≡ 1 (mod 12), write α = 12l + 1 for some integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, α ≡ 1 (mod 3) ⇒ α + 2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Substituting α = 12l + 1 in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='7, we get Y 2 − � (12l + 1)2 − 4 � Z2 = aαd − b, ⇒ Y 2 ≡ aαd − b (mod 3), ⇒ Y 2 ≡ a (12l + 1)d − a2d + 3 (mod 3), ⇒ Y ≡ 1 − 2d (mod 3), ⇒ Y 2 ≡ 2 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' A contradiction as 2 is not square modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Sub-case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If α ≡ 9 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, there is a prime factor p ≡ 5 or 7 (mod 12) of (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let p ≡ 5 or 7 (mod 12) be a prime factor of (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus, Y 2 ≡ aαd − b (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let α = pl + 2 for some integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, Y 2 ≡ a (pl + 2)d − b (mod p), ⇒ Y 2 ≡ 3 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' This leads to a contradiction as 3 is not a quadratic residue modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Sub-case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' When α ≡ 5 (mod 12), we substitute α = 3k + 2 for some integer k and get, Y 2 − � (3l + 2)2 − 4 � Z2 = (12l + 1) (3k + 2) − 2d (12l + 1) + 3, ⇒ Y 2 ≡ 2 − 2d ≡ 0 (mod 3), ⇒ Y ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 6 OM PRAKASH AND KALYAN CHAKRABORTY Further, we substitute Y = 3m and α = 12n + 5 for some integers n and m in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='7 and arrive onto, 9m2 − (12n + 3) (12n + 7) Z2 = a (12n + 5)d − b, ⇒ − (n + 1) Z2 ≡ d−1 � i=0 (12n + 5)d−1−i 2i (mod 3), ⇒ − (n + 1) Z2 ≡ 1 (mod 3), ⇒ n ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Hence, α ≡ 17 (mod 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Note that 3 divides (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus there is a prime factor p ≡ 5 or 7 (mod 12) of (α−2) 3 , otherwise it would mean that α−2 3 is congruent to ±1, which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Therefore, α − 2 ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus, Y 2 ≡ aαd − b (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Substituting α = pl + 2 for some integer l, we have Y 2 ≡ 3 (mod p), which contradicts the fact that 3 is quadratic residue modulo p if p ≡ ±1 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' □ Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The result of Sury and Majumdar [5] follows by substituting a = 1 and d = 3 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The particular case d = 3 in the same theorem deduces the results of Vaishya and Sharma [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' By increasing the exponents in the expression for b to 3, we will now examine the Diophantine equation with a little more generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The potential of a solution in this scenario is described by the following two corollaries, along with a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The equation axd − y2 − z2 + xyz − b = 0 has no integer solution (x, y, z) with x even for fixed integers a and b such that a ≡ 1 (mod 12) and b = 2da − 3r with positive odd integers r and d as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We follow exactly the same steps as in Case 1 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Suppose there is a solution with x = α even, then we write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2) as: � y − αz 2 �2 − �α2 4 − 1 � z2 = aαd − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='8) Let Y = y − αz 2 , β = α 2 and z = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='8) can be written as, Y 2 − � β2 − 1 � Z2 = aαd − b = 2dβda − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='9) If β is even, say β = 2n for some integer n, then the reduction modulo 4 of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='9) will give, Y 2 + Z2 ≡ 3r ≡ 3 (mod 4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='10) which is not feasible in Z/4Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If β is odd, say β = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, the reduction modulo 4 of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='9) provides, Y 2 ≡ 3r ≡ 3 (mod 4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='11) which again is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' □ The following corollary deals with solutions having x, an odd integer: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The equation axd − y2 − z2 + xyz − b = 0 has no integer solution in x, y and z with x ≡ 1 or 9 (mod 12), for fixed integers a, b such that a ≡ 1 (mod 12) and b = 2da − 3r, for r and d as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Analogous steps as in Sub-case 2 and 3 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1 will give the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2 says that, if there is a solution of axd − y2 − z2 + xyz − b = 0 with a and b as described in the Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2, then x must be 5 modulo 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We will see some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For a = 25, d = 3 and r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The equation 25x3 − y2 − z2 + xyz − 173 = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='12) has no integer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 8 OM PRAKASH AND KALYAN CHAKRABORTY Example 2 shows that the equation may not have solution even with x ≡ 5 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' However, the next examples tell us the other possibility as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If a = 13, d = 3 and r = 3, then 13x3 − y2 − z2 + xyz − 77 = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='13) has an integer solution (5, = 18, −102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The condition that r should be odd is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For a = 13, d = 3 and r = 2, the equation 13x3 − y2 − z2 + xyz − 95 = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='14) has an integer solution (2, −10, −7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Hyperelliptic curves A hyperelliptic curve H over Q is a smooth projective curve associated to an affine plane curve given by the equation y2 = f (x), where f is a square-free polynomial of degree at least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If the degree of f is 2g + 1 or 2g + 2, then the curve has genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We write H (Q) for the set of Q-points on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Determining rational points on hyperelliptic curve is one of the major problems in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The following is the general result regarding the size of H (Q), which was conjectured by Mordell and was proved by Faltings: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [2] If C is a smooth, projective and absolutely irreducible curve over Q of genus at least 2, then C (Q) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' We may thus, at least theoretically, write down the finite set C (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' It is still a signifi- cant unresolved problem to perform this practically for a given curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Given a hyperelliptic curve H, we can define the height (classical) function to be the maximum of absolute values of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The Northcott property tells us that there are finitely many equations with bounded height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Thus, one may talk about the density and averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In this regard, Bhargava [1] has proved that most of the hyperelliptic curve over Q has no rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' So, most of the times calculating H (Q) means proving H (Q) = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' In this section, we construct hyperelliptic curves corresponding to the equation axd − y2 − z2 + xyz − b = 0 with a and b as mentioned in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then, we prove that H (Q) = φ (corroborating Bhargava [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' The main ingredient to prove this is the following Nagell-Lutz type theorem (Theorem 3, [3]) proved by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES 9 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [3] Let C be a nonsingular projective curve of genus g ≥ 1 given by y2 = x2g+1 + b1x2g + · · · + b2gx + b2g+1, where bi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Suppose ψ : C (Q) → J (Q) be the Abel-Jacobi map, defined by ψ (p) = [p − ∞], where J (Q) is the Jacobian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' If p = (x, y) ∈ C (Q) \\ {∞} and ψ (p) ∈ J (Q)tors, then, x, y ∈ Z and either y = 0 or y2 divides discriminant of the polynomial x2g+1 + b1x2g + · · · + b2gx + b2g+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For fixed m we define hyperelliptic curves, Hm,a,b : y2 − mxy = axd − m2 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Suppose m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Then write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1) as: � y − mx 2 �2 − m2x2 4 = axd − m2 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1) Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1) by ad−1 throughout, and using the fact that d is odd and divisible by 3, we have, �� y − mx 2 � a d−1 2 �2 − ad−1m2x2 4 = (ax)d − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2) We get the following hyperelliptic curve by substituting �� y − mx 2 � a d−1 2 � = Y and ax = X, He m,a,b : Y 2 − ad−3m2X2 4 = Xd − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3) Now if m is odd, multiply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2) by 4d throughout to get �� y − mx 2 � a d−1 2 2d�2 − (4a)d−1 m2x2 = (4ax)d − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Finally substitute �� y − mx 2 � a d−1 2 2d� = Y and 4ax = X, to get Ho m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='4) Let, Hm,a,b = \uf8f1 \uf8f2 \uf8f3 He m,a,b if m is even Ho m,a,b if m is odd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='5) be the hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let a and b be as defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For any m ∈ N, the hyperelliptic curve Hm,a,b has torsion-free Mordell-Weil group over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 10 OM PRAKASH AND KALYAN CHAKRABORTY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Let a and b be fixed positive integers with a ≡ 1 (mod 12) and b = 2da − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For any even integer m, consider the hyperelliptic curve He m,a,b : Y 2 − ad−3m2X2 4 = Xd − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='6) By Theorem 3 of [3], if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='6) has an integer solution (X0, Y0), then � aX0, �� Y0 − mX0 2 � a d−1 2 � , m � is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' However, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1 we have proved that it has no integer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' For an odd integer m, consider the hyperelliptic curve Ho m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='7) Suppose (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='7) has a solution (X0, Y0), then � 4aX0, �� Y0 − mX0 2 � a d−1 2 2d� , m � is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='1), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Numerical examples In this section we give some numerical examples corroborating our results in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='2 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' a d r Equation Solution 1 3 3 x3 − y2 − z2 + xyz + 19 = 0 (5, 0, −12) 1 3 5 x3 − y2 − z2 + xyz + 235 = 0 (29, 12, −60) 1 3 7 x3 − y2 − z2 + xyz + 2179 = 0 (5, 0, −48) 1 3 9 x3 − y2 − z2 + xyz + 19675 = 0 (−31, 12, −30) 13 3 3 13x3 − y2 − z2 + xyz − 77 = 0 (5, −18, −102) 13 3 5 13x3 − y2 − z2 + xyz + 139 = 0 (5, 0, −42) 13 3 7 13x3 − y2 − z2 + xyz + 2083 = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' 25 3 3 25x3 − y2 − z2 + xyz − 173 = 0 (5, 0, −42) Acknowledgement This work is done during the first author’s visit to Institute of Mathematical Sci- ences (IMSc), Chennai, and he is grateful to the Institute for the hospitality and the wonderful working ambience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Both the authors are grateful to Kerala School of Mathe- matics(KSoM), Kozhikode, for it’s support and wonderful ambience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES 11 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Bhargava, Most hyperelliptic curve over Q have no rational point, arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='0395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Faltings, “Finiteness theorems for abelian varieties over number fields”, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=', 73 (1983), 349–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Grant, On an analogue of the Lutz-Nagell theorem for hyperelliptic curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Number Theory, 133 (2013), 963–969.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Vaishya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Sharma, A class of fruit Diophantine equations, Monatshefte f¨ur Mathematik, 199 (2022), 899–907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Kerala School of Mathematics, Kozhikode - 673571, Kerala, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Email address: omprakash@ksom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='in Kerala School of Mathematics, Kozhikode - 673571, Kerala, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content=' Email address: kalychak@ksom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'} diff --git a/MNE3T4oBgHgl3EQfBAmd/vector_store/index.faiss b/MNE3T4oBgHgl3EQfBAmd/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..16c84d28fad52f8ee1afed23897c7731df959fb2 --- 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Current Constraints on Antineutrino Production by 241Pu and Paths Towards the +Precision Reactor Flux Era +Yoshi Fujikake, Bryce Littlejohn,* and Ohana B. Rodrigues† +Physics Department, Illinois Institute of Technology, Chicago, IL 60616, USA +Pranava Teja Surukuchi‡ +Wright Laboratory, Yale University, New Haven, CT 06520, USA +By performing global fits to inverse beta decay (IBD) yield measurements from existing neutrino experi- +ments based at highly 235U enriched reactor cores and conventional low-enriched cores, we explore current +direct bounds on neutrino production by the sub-dominant fission isotope 241Pu. For this nuclide, we determine +an IBD yield of σ241 = 8.16 ± 3.47 cm2/fission, a value (135 ± 58)% that of current beta conversion models. +This constraint is shown to derive from the non-linear relationship between burn-in of 241Pu and 239Pu in con- +ventional reactor fuel. By considering new hypothetical neutrino measurements at high-enriched, low-enriched, +mixed-oxide, and fast reactor facilities, we investigate the feasible limits of future knowledge of IBD yields +for 235U, 238U, 239Pu, 241Pu, and 240Pu. We find that first direct measurement of the 240Pu IBD yield can be +performed at plutonium-burning fast reactors, while a suite of correlated measurements at multiple reactor types +can achieve a precision in direct 238U, 239Pu, and 241Pu yield knowledge that meets or exceeds that of current +theoretical predictions. +I. +INTRODUCTION +A nuclear reactor primarily generates thermal energy as +product nuclei inherit (as kinetic energy) and deposit (through +repeated elastic collisions) excess rest mass energy from the +fission of heavy nuclides in the reactor’s fuel, such as 235U, +238U, 239Pu, 241Pu, and more. Successive decays of these +neutron-rich product nuclei release additional energy in the +form of beta particles, gamma-rays, and antineutrinos. While +the two former product types are additional, sub-dominant +contributors to heat generation in a reactor, the antineutri- +nos (νe ) and their associated kinetic energy entirely escape +the reactor core, offering an attractive avenue for studying +the properties of neutrinos [1–3], interrogating state-of-the- +art nuclear data [4], and non-intrusively monitoring nuclear +reactor cores [5]. Reactor-based νe detectors have demon- +strated that neutrinos have mass [6–9], and have searched for +the existence of new heavy neutrino states [10–15] and other +new physics phenomena [16–23]. By observing discrepan- +cies with respect to existing theoretical νe flux and energy +spectrum predictions, they have also highlighted limitations of +and/or inaccuracies in community fission yield and beta decay +databases [7, 9, 24–32]. Antineutrino monitoring case studies +have explored a variety of potential use case scenarios, such +as thermal power load-following and determination of reactor +fissile inventory [33–37], and existing νe detectors have con- +firmed the feasibility of some of these activities [25, 38, 39]. +The average number of antineutrinos released or detected +per nuclear fission depends on the fission isotope in question: +different fission isotopes have different fission product yields, +with each product varying in its distance from the line of sta- +bility and having its own unique nuclear structure and decay +* blittlej@iit.edu +† obenevidesrodrigues@iit.edu +‡ pranavateja.surukuchi@yale.edu +scheme. Thus, reactor cores with differing fuel compositions +are expected to differ in their rate of νe output. These ex- +pected differences have been explicitly demonstrated in recent +νe experiments using the inverse beta decay (IBD) interaction +process, p + νe → e++ n, which has a 1.8 MeV interaction +threshold and a precisely-predicted cross-section, σIBD(Eν), +versus νe energy Eν [40]. For these experiments, measured +νe fluxes have been expressed in terms of an IBD yield per +fission σf [1]: +σf(t) = +� +i +Fi(t)σi. +(1) +In this expression, Fi(t) is the fraction of fissions contributed +by isotope i in the sampled reactor core(s) during the experi- +ment’s measurement period and σi its IBD yield per fission, +σi = +� +Si(Eν)σIBD(Eν)dEν. +(2) +Here, Si(Eν) is the true produced νe energy (Eν) spectrum +per fission for isotope i, and σIBD is the inverse beta decay +interaction cross-section. +In a straightforward demonstration of variations in νe emis- +sion between fission isotopes, reported IBD yields σf are +clearly offset [41] between measurements at 235U-burning +highly enriched reactor cores [42–48] and measurements per- +formed at commercial cores burning a mixture of 235U, 238U, +239Pu, and 241Pu [7, 49–56]. In a separate demonstration, the +Daya Bay and RENO experiments have compared IBD yields +measured in the same detectors at differing points in their sam- +pled commercial reactors’ fuel cycles, observing higher yields +during periods with higher (lower) 235U (239Pu) fission frac- +tions [25, 55]. +By performing fits to a set of σf measurements at reactors +of well-known fission fraction Fi, one can use Equations 1 +and 2 to directly determine the isotopic IBD yield σi of one +or more fission isotopes. With a single HEU-based experi- +ment, the IBD yield for 235U, σ235, can be trivially deter- +mined as σ235 = σf, since F235 approaches unity for these +arXiv:2301.13123v1 [hep-ph] 30 Jan 2023 + +2 +cores. On their own, HEU-based σ235 measurements exhibit +deficits [57] with respect to commonly-used beta-conversion +predictions [58, 59], indicating issues in modeling either the +core’s νe emissions or νe behavior during propagation [60]. +Daya Bay and RENO σf measurements, which encompass +multiple data points with differing LEU fuel composition +F235, 238, 239, 241, when combined with modest theoretical +constraints on σ238, 241, yields from the sub-dominant iso- +topes 238U and 241Pu, enable determination of isotopic yields +for both 235U and 239Pu [25, 55]. These measurements show +a deficit with respect to 235U conversion predictions, but no +such deficit for 239Pu, providing further credence to the νe +emission mis-modelling hypothesis. Going further, global fits +of both LEU and HEU datasets can be used to simultaneously +determine σ235, 238, 239 [61]: the measured σ238 shows a sig- +nificant (33±14)% deficit [62] with respect to summation pre- +dictions based on community-standard nuclear databases [59], +suggesting potential issues in current 238U fission yield mea- +surements or evaluations. Future direct determination of iso- +topic IBD yields for a wider array of fission isotopes beyond +235U, 239Pu, and 238U, as well as improved precision for these +three isotopes, can lead to further understanding or improve- +ment of existing nuclear data, reactor νe models, and reactor- +based fundamental physics studies. +Improved isotopic IBD yield measurements also hold po- +tential benefits for future νe -based applications. Some ad- +vanced reactor technologies present unique safeguards chal- +lenges that may be satisfied by near-field νe -based monitor- +ing capabilities [63]. However, neutrino emissions have never +been measured at advanced reactor cores, some of which dif- +fer substantially from measured HEU and LEU reactor types +in both fuel composition and core neutronics [35, 64, 65]. +For example, mixed-oxide reactor fuels, which, unlike con- +ventional low-enriched fuel, are produced from a mixture of +uranium and plutonium isotopes, may be deployed in future +reactors to realize a closed nuclear fuel cycle or as a means of +disposing of existing plutonium stockpiles. Fast fission reac- +tor technologies, which, unlike conventional thermal reactors, +rely on fast neutron induced fission to maintain criticality, may +offer safety and sustainability advantages with respect to con- +ventional reactor types. For these reactors, better direct deter- +minations of true underlying σi can enable more robust and +reliable future monitoring capabilities than would be possible +using existing demonstrably imperfect models of νe produc- +tion per fission. +In this paper, we study how existing and potential future +IBD measurements can provide first direct glimpses at νe +production by previously unexplored fission isotopes and im- +prove our precision in understanding of the more-studied iso- +topes 235U, 239Pu, and 238U. By performing loosely con- +strained fits of isotopic IBD yields to existing LEU and HEU +datasets, we demonstrate the feasibility of achieving non- +trivial future bounds on νe production by 241Pu. By applying +the same fit techniques to hypothetical future high-precision +IBD yield measurements at HEU, LEU, MOX, and fast fis- +sion reactors, we show that direct IBD yield determinations +for all four primary fission isotopes (235U, 238U, 239Pu, and +241Pu) can meet or exceed the claimed precision of exist- +ing conversion-based predictions while also placing the first +meaningful bounds on 240Pu νe production. +We begin in Section II with a description of the global fit +and existing and hypothetical future IBD yield datasets. Re- +sults of the fit to existing datasets and studies of 241Pu limits +are presented and discussed in Sections III. In Section IV, we +describe the set of considered future hypothetical experiments +and the result of applying global fits to the hypothetical re- +sults of these experiments. Main results are then summarized +in Section V. +II. +GLOBAL DATASETS AND FIT TECHNIQUE +In this analysis we perform fits to a set of IBD rate measure- +ments with varying degrees of systematic correlation between +each measurement set. For an individual measurement, the +number of IBD interactions N detected per time interval t can +be described as: +N = NpεP(L) +4πL2 +� Wth(t)σf(t) +¯E(t) +dt, +(3) +where Np is the number of target protons, ε is the efficiency +of detecting IBDs, P(L) is the survival probability due to +neutrino oscillations, and L is the core-detector distance. Of +the time-dependent quantities, Wth(t) is the reactor’s thermal +power, ¯E(t) = � +i Fi(t)ei is the core’s average energy re- +leased per fission, ei is the average energy released per fission +of isotope i, and Fi(t) and σf(t), as in Equations 1 and 2, are +the fission yields and IBD yields of isotope i. In order to per- +form one or multiple measurements of σf, a reactor νe flux +experiment must measure N while characterizing the other +reactor and detector inputs in Equation 3. +A. +Existing Datasets +Many experiments have successfully measured σf values +and associated statistical and systematic uncertainties. As in- +put for this study, we include time-integrated IBD yield mea- +surements and uncertainties reported by the Goesgen, Bugey- +3, Bugey-4, Rovno, Palo Verde, CHOOZ, and Double Chooz +LEU-based experiments and the ILL, Savannah River, Kras- +noyarsk, Nucifer and STEREO HEU-based experiments, as +well as the highly-correlated datasets at varying Fi from the +Daya Bay and RENO experiments. Calculated fission frac- +tions and measured yields for these experiments, as well as as- +sociated uncertainties and cross-measurement systematic cor- +relations, have been summarized in Ref. [66], and are used +for portions of this paper’s analysis. Input data tables are pro- +vided in the public GitHub repository [67] provided by the +authors as an accompaniment to this analysis. Since we do +not consider short-baseline oscillations as part of this analysis, +reactor-detector baselines are not used in analysis of existing +datasets, but are nonetheless provided in these tables. + +3 +B. +Hypothetical Future Datasets +For this study, we also generate hypothetical future IBD +yield datasets and uncertainty budgets matching the expected +capabilities of experimental deployments at HEU, LEU, MOX +and fast reactor types. These are also provided in the GitHub +supplementary materials, along with assumed uncertainty co- +variance matrices for all considered hypothetical measure- +ments. +Hypothetical IBD yield measurements are Asimov +datasets free of statistical and systematic fluctuations that are +generated according to Equation 3. As input to this equa- +tion, fission fractions are required for each host reactor and +are described below. +To match general indications from +recent summations [32] and fission beta [68], and νe flux +evolution [25] measurements, and matching the approach in +Ref. [61], we choose input ‘true’ IBD yield values match- +ing a scenario where Huber-Mueller modelled yields [69] +are only incorrect for 235U: (σ235, σ238, σ239, σ241) = +(6.05,10.10,4.40,6.03) ×10−43 cm2/fission. +The yield for +240Pu has not been predicted in the literature to our knowl- +edge, so we estimate it by applying a 3Z-A scaling suggested +in Ref. [70] to the four previously-mentioned isotopes; the +determined central value is σ240 = 4.96 ×10−43 cm2/fission. +Other experimental assumptions regarding detector, reactor, +and experimental layout parameters are then required to de- +fine the statistical and systematic uncertainties associated with +each hypothetical IBD yield measurement. +The HEU-based measurement is modeled after the HFIR +facility at Oak Ridge National Laboratory, sporting 85 MW +of thermal power, a 100% 235U fission fraction, and a 7 m +reactor-detector center-to-center distance. LEU-based mea- +surements are assumed to occur at a 20 m center-to-center dis- +tance from a core following the attributes of a 2.9 GWth Daya +Bay core with an 18 month fuel cycle. Assumed fission frac- +tions are chosen to fall roughly in the middle of the range +reported for Daya Bay’s cores in Fig. 1 of Ref. [75], and cor- +respond to a fully-loaded core with roughly 1/3 of its rods con- +taining fresh (pure uranium oxide at start-up) fuel; this level +of partial reloading is customary when operating cores of this +type. +MOX-based measurements are modelled after the MOX +reactor studies of Ref. [72], and is assumed to occur 20 m +from a core with a 3.2 GWth thermal power and 18 month cy- +cle length, and fission fractions matching those of the sim- +ulated 50% weapons-grade MOX-burning core. +Weapons- +grade (WG) plutonium is characterized by low 240Pu and +241Pu isotopic fractions, and thus a low F241 fission fraction +at reactor start-up. These WG-MOX core parameters corre- +spond to a realizable operational scenario implemented for +the goal of plutonium stockpile disposition in a commercial +reactor core. We will also reference a similar case where 50% +reactor-grade (RG) MOX fuel is used in the same reactor type; +these parameters correspond to an operational scenario for a +commercial complex operated as part of a closed nuclear fuel +cycle program. Following recommendations of the authors +of Ref. [72], fission fractions for the WG-MOX core exam- +ple are assumed to match the reported fission fractions for the +first third of pictured 50% WG-MOX running in Ref. [72], +while fractions from the RG-MOX case are assumed to match +the those of 50% WG-MOX running between days 800 and +1350 [76]; fission fractions were extracted by interpolating +fission rates from this reference and normalizing such that the +sum for the four primary fission isotopes is equal to unity. +It should be stressed that modeled fuel content evolution for +LEU and MOX cores is highly dependent on the initial condi- +tions of the fuel, on the neutronics of the involved core type, +and on reactor operations. In this study, we include one spe- +cific fission fraction set for each fuel type – LEU, WG-MOX, +and RG-MOX; the impact or potential benefits of further vari- +ations between LEU or MOX core types is not considered. +Finally, two experiments are assumed to occur at the base- +lines of 20 m and 7 m distances from the primarily plutonium- +burning 1.25 GWth PFBR fast breeder reactor in India [73] +and the 300 GWth Versatile Test Reactor fast reactor [74] re- +spectively. The former reactor plays a central role in plans for +realization of an independent, sustainable nuclear fuel cycle +in India, while the latter has been developed as a US-based +reactor materials and irradiation R&D facility based at Idaho +National Laboratory [64]. Assumed reactor and site parame- +ters for all measurements are summarized in Table I; fission +fraction values for all hypothetical measurement data points +used in this study are illustrated in Figure 1. +0 +5 +10 +15 +20 +25 +30 +Data Points +0 +0.2 +0.4 +0.6 +0.8 +1 +Fission Fraction +Fission fractions in PROS HEU+LEU+WGMOX+RGMOX+VTRRx+PFBR +U235 +U238 +Pu239 +Pu240 +Pu241 +HEU +LEU +WG MOX +RG MOX +VTR +PFBR +FIG. 1. Fission fractions used for hypothetical future measurement +data points described in this section. See text for details. +For all experiments, an IBD detector matching qualities of +the 4 ton PROSPECT reactor νe detector are used [77]; rel- +evant parameters are also listed in Table I. In some cases, a +1 ton detector with otherwise similar experimental parame- +ters is also considered; this case enables investigation of the +value of using a near-future compact νe monitoring detector, +such as the Mobile Antineutrino Demonstrator [78] (MAD), +to perform IBD yield benchmarking measurements at multi- +ple reactor locations. In all cases, the statistical uncertain- +ties associated with each datapoint for each reactor-detector +combination are estimated using the associated detector and +reactor parameters quoted in Table. I and lie between 0.15 +% and 0.2 %. For simplicity, we do not consider statistical + +4 +Parameter +HEU LEU MOX Fast (PFBR) Fast (VTR) +Reactor +Thermal Power (MWth) +85 +2900 3200 +1250 +300 +Burnup Profile +- +[71] +[72] +[73] +[74] +Reactor Cycle Length +24 d +1.5 y 1.5 y +1.5 y +100 d +Experimental +Core-Detector Distance (m) +7 m +20 m 20 m +20 m +20 m +Data-Taking Length +3 y +1.5 y 1.5 y +1 y +100 d +Detector +Active Mass +4 ton (1 ton) +Target Protons +2×1029 (0.5×1029) +IBD Detection Efficiency +40% +Uncertainty, Reactor +Thermal Power +1.0% 0.5% 0.5% +1.0% +1.0% +Fission Fractions +- +0.6% 0.6% +0.6% +0.6% +Energy per Fission +0.1% 0.2% 0.2% +0.2% +0.2% +Uncertainty, Detector +Target Protons +1.0% +Detection Efficiency +0.75% +IBD Cross Section +0.1% +Total Reactor Systematic +0.5% 0.8% 0.8% +1.2% +1.2% +Total Detector Systematic +1.3% +TABLE I. Assumed reactor and site parameters for the hypothetical future short-baseline reactor experiments described in the text. +and systematic uncertainty contributions from IBD-like back- +grounds; for a PROSPECT-like detector expecting signal-to- +background ratios of better than 4 (10) deployed on-surface at +an HEU (LEU) reactor [79], IBD counts would be expected to +dominate measurement statistical uncertainties. +Hypothetical measurements should also be accompanied by +predicted reactor- and detector-related systematic uncertain- +ties, which are also summarized in Table I. Systematics for +most cores are dominated by the uncertainty in the thermal +power produced by the operating core. Commercial reactor +companies have provided sub-percent precision in reported +thermal powers for existing IBD yield measurements at nu- +merous reactor sites [80, 81]; for this reason, we choose 0.5% +uncertainty for LEU and MOX cores. While similar thermal +power measurement devices and strategies could be applied +to HEU facilities, in practice, legacy systems used in exist- +ing HEU facilities have recently provided thermal power un- +certainties closer to 2% [48]; for this analysis, we optimisti- +cally assume implementation of upgraded measurement sys- +tems or techniques capable of providing 1% precision at an +HEU core. Advanced technologies for time-stable and high- +precision thermal power monitoring in sodium-cooled fast re- +actors like PFBR and VTR are under active development, due +to the difficulties associated with the coolant’s high temper- +ature and chemical corrosiveness; given the lack of available +quantitatively demonstrated capabilities, we also assume a 1% +thermal power uncertainty for this core type. While thermal +power uncertainties for different reactors are assumed to be +uncorrelated, this uncertainty is correlated between multiple +measurements at the same core. Measured IBD yields for an +experiment will also be uncertain due to the limits in knowl- +edge of fission fractions in the core, which is defined via de- +tailed reactor core simulations. In the absence of these cal- +culations for all core types, we will assume an uncertianty +of 0% for the HEU experiment and 0.6% for all other cores, +following the value quoted by Daya Bay and others for LEU +cores [71]. This uncertainty is also assumed to be uncorre- +lated between cores, but correlated between measurements at +the same core. Isotopic energy release per fission ei – re- +quired for calculating expected experiment statistics – have +minor IBD yield uncertainty contributions of 0.1% to 0.2% +depending on core fuel content [82]; the ei central value and +uncertainty for 240Pu is assumed to match that of 241Pu. +On the detector side, uncertainties are dominated by the +limited knowledge of IBD detection efficiency, assumed to be +known with 0.75% precision, as well as knowledge of the to- +tal number of protons within the detector’s target region, as- +sumed to be known to 1%; these chosen values reflect those +achieved in a range of recent large- and small-detector IBD +experiments [48, 55, 83, 84]. In this analysis, we consider +the possibility of moving a single reactor neutrino detector to +multiple reactor core types to perform systematically corre- +lated IBD yield measurements; for this reason, unless other- +wise mentioned, we treat detector systematic uncertainties as +correlated between all measurements. + +5 +C. +Global Fit Approach +To obtain isotopic IBD yields in this analysis, we use a +least-squares test statistic: +χ2 = +� +a,b +� +σf,a − r +� +i +Fi,aσi +� +V−1 +ab +� +σf,b − r +� +i +Fi,bσi +� ++ +� +j,k +(σth +j − σj)V−1 +ext,jk(σth +k − σk). +(4) +In this fit, experimental inputs Fi and σf are as described +above, and the sum i is run over five fission isotopes, 235U, +238U, 239Pu, 241Pu, and 240Pu, with five attendant IBD yield +fit parameters. +The experimental covariance matrix V de- +fines the uncertainties for each experiment and their cross- +correlations, as described in the previous-subsection. +The +final term is used to constrain fitted σi values to theoreti- +cal predictions by adding a penalty that increases as the two +quantities diverge. +In contrast to most recent global IBD +yield fits [25, 57, 85], we are interested in examining weakly- +constrained or un-constrained simultaneous fits of all relevant +fission isotopes’ IBD yields. For this reason, j and k sum only +over the three sub-dominant isotopes, 238U, 241Pu, and 240Pu, +and the 3×3 V−1 +ext is diagonal (no assumed uncertainty correla- +tion between isotopes), with elements set to achieve wide 1σ +theoretical constraints of 75% of the predicted yield. To com- +pare to previous IBD yield fits [61, 86], we occasionally con- +sider the much tighter (2.6%) bounds on σ241 quoted by the +Huber model [58]. For fits not involving fast reactor datasets, +σ240 is pegged to the theoretically-predicted value, and has no +effect on the subsequent 4-parameter fit. +III. +FITS TO EXISTING DATASETS AND 241PU IBD +YIELD CONSTRAINTS +We first consider IBD yield fits applied to the existing +global yield datasets described briefly in Section II A. By first +applying tight 2.6% constraint on 241Pu, we largely reproduce +unconstrained 235U, 238U, and 239Pu yield best-fit values re- +ported for the oscillation-free fit in Ref. [86]. Test statistic +values with respect to the best-fit (∆χ2) versus input value +are shown for each isotope in Figure 2, while minimizing over +the three other isotopic yield parameters. We observe a best- +fit 235U yield more than 3σ (5%) below the Huber-predicted +value, and a best-fit 238U yield that deviates from the pre- +dicted central value by (36±20), slightly more than in previ- +ous fits [86]. As in previous fits, the 239Pu yield is found to be +consistent with Huber-predicted values within a 5%, ∼ 1σ un- +certainty band. This similarity in results indicates that the rel- +atively new STEREO data point [48], while qualitatively bol- +stering confidence in historical observations of a ∼5% yield +deficit at HEU cores [87], has fairly modest quantitative im- +pact on the primary issues surrounding data-model agreement +for conversion-predicted uranium IBD yields. +With consistency established with respect to previous anal- +yses, we proceed with loosening of yield constraints for all +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.1 1.2 +i +R +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +2 +χ + +∆ + 0.013 +± + = 0.953 +235 +σ + 0.201 +± + = 0.642 +238 +σ + 0.044 +± + = 1.047 +239 +σ + 0.026 +± + = 1.001 +241 +σ +0.6 +0.8 +1 +1.2 +1.4 +1.6 +i +R +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +2 +χ + +∆ + 0.013 +± + = 0.953 +235 +σ + 0.264 +± + = 0.730 +238 +σ + 0.252 +± + = 0.910 +239 +σ + 0.426 +± + = 1.353 +241 +σ +FIG. 2. Isotopic IBD yield fit results for the existing global dataset +with tight (top, 2.6%) and loose (bottom, 75%) external constraints +on the 241Pu yield. Test statistic values with respect to the best-fit +(∆χ2) are shown versus input value for each of the four primary +fission isotopes. For each isotope’s curve, the fit is marginalized over +the other isotopes. +fission isotopes. Figure 2 shows reported isotopic ∆χ2 test +statistic values versus input σ value for each isotope while +applying a looser constraint on 241Pu of 75%. Best-fit param- +eters and 1σ ranges are found to be: +σ235 = 6.37 ± 0.08; +σ238 = 7.37 ± 1.95; +σ239 = 4.00 ± 1.01; +σ241 = 8.16 ± 3.47. +(5) +The best-fit χ2 +min is found to be 26.2 for 38 degrees of free- +dom (41 data points, 3 fit parameters), indicating an accept- + +6 +able goodness-of-fit. +However, this value is only slightly +lower than that provided by the more-constrained fit (χ2 +min += 26.6), indicating that this enhanced freedom has not sub- +stantially improved data-model agreement. +Central values +of 235U, 238U, and 239Pu fit parameters are relatively sta- +ble, remaining within 15% of those provided by the more- +constrained fit. Meanwhile, the newly freed 241Pu yield in- +creases by 35%, although σ241 nonetheless remains consistent +with its model-predicted value within its large 43% relative +uncertainty band. Thus it appears that the current global IBD +yield dataset does not have the statistical power to provide +meaningful tests of underlying modelling issues for 241Pu. +The disappointing lack of new insight should not be too sur- +prising, given the small (O(5%) or less) fractional contribu- +tion of 241Pu fissions in all existing measured reactor cores. +However, it is interesting to note that σ241 1σ error bands +are found to be tighter than the externally-applied constraint. +This indicates that there are features in the existing global +dataset that provide the power to specifically constrain 241Pu. +To attempt to identify these features, we examined correla- +tions between fitted isotopic yields, which are depicted in Fig- +ure 3 as best-fit parameter space regions in two dimensions be- +tween 241Pu and the other three isotopes. Substantial 241Pu- +239Pu and 241Pu-238U degeneracies can be observed, with the +former reflected in a more than five-fold increase in uncertain- +ties on σ239 between the more-constrained (4% uncertainty) +and less-constrained (25% uncertainty) fits. Degeneracies can +also be expressed by calculating correlation coefficients be- +tween the fitted yield parameters, which are also given in the +legends of Fig 3: +ρσi,σj = (σi − σi)(σj − σi) +σσiσσj +(6) +The extreme 241Pu-239Pu correlation can be understood by +observing the fission fraction evolution trends experienced by +LEU reactors, as depicted in Figure 1. In these cores, F239 +and F241 rise in tandem with reactor fuel burn-up, making it +hard for unconstrained fits to simultaneously determine both +σ239 and σ241. It can also be understood as a simple reality +of underlying nuclear physics in the core: 241Pu is produced +by via multi-neutron capture on 239Pu, and thus its build-up in +the core is dependent on the build-up of the latter. In aspects +of previous multi-datapoint LEU analyses, such as those of +Daya Bay [25, 88] and RENO [55], 241Pu and 239Pu fission +fractions are treated as explicitly linearly correlated. +We examine the limits of this linear correlation by gener- +ating hypothetical LEU reactor IBD yield datasets following +the method described in Section II B and fission yields from +Figure 1: one dataset assumes isotopic yields matching the +best-fit for the existing global dataset, and the other assumes +true 241Pu and 239Pu yields close to the axis of anti-correlation +between the two datasets, but beyond the 1σ bounds allowed +by the data. Chosen true yields for this test are illustrated +in the right panel of Figure 3; the 238U yield for this case, +8.8 cm2/fission, was chosen to vertically align the two yield +datasets for easier comparison of trends. Hypothetical yields +for these two cases are pictured in Figure 4. The test cases +clearly differ in the change in slope, or curvature, present in +the LEU data points, providing an indication of the primary +source of unique 241Pu yield information in current and fu- +ture experimental data. The extreme 239Pu-241Pu yield offset +in this example amplifies the impact of a modest non-constant +relationship between F239 and F241 in LEU-based datasets, +which is also illustrated in Figure 4. To test the validity of +this hypothesis with existing datasets, we perform a fit to only +the RENO and Daya Bay LEU datasets while applying loose +75% external constraints on all four isotopes. While large un- +certainty increases are seen in σ235 and σ238, σ239 and σ241 +fractional uncertainties are altered by <30%, and fractional +bounds on σ241 (43%) remain tighter than the 75% external +constraint. Thus, in the existing global dataset, it does ap- +pear that that the Daya Bay and RENO LEU data points are +responsible for the modest breaking of degeneracy between +239Pu and 241Pu yields. +Adding this to previously-established trends, it is straight- +forward to recount the independent features of the global IBD +yield dataset that enable determination of all four isotopes’ +IBD yields: +• HEU-based experiments’ σf measurements directly +constrain σ235 [57]. +• The measured relative linear σf slope versus fuel +burn-up at LEU-based experiments directly constrains +σ239 [25]. +• The time-integrated offset in σf between HEU and LEU +cores constrains σ238 [61]. +• The curvature of σf slope versus fuel burn-up at LEU +experiments constrains σ241. +As we move on to consider possible future IBD yield mea- +surement scenarios, these high-level principles serve to guide +attention toward those with particular promise for improving +global knowledge of isotopic yields. In particular, we will +look to explore new multi-dataset measurements that can pro- +vide an enhanced view of σf curvature with host reactor fuel +evolution. +We end this Section by noting that within the current global +dataset, Daya Bay contains currently-unexploited potential. +Ref. [75] indicates O(5%) F241/F239 variations between re- +actor cycles that are averaged out in its current fuel content +binning scheme. To estimate the achievable gains in the fis- +sion yields, we generate an Asimov IBD yield dataset with fis- +sion fractions taken from a combination of rates, RENO and +Daya Bay-like experiment is divided into two halves; one with +the default fission fractions while the other having F241/F239 +relatively reduced by 2.5%. The systematic and statistical un- +certainties are assumed to match the existing global dataset +and the yields are generated using best-fit results from the +global dataset. Such a joint fit provides a modest improvement +in the precision of fission yield of (σ235, σ238, σ239, σ241) = +(1.3%, 24.8%, 19.7%, 39.2%) compared to the precision of +(1.3%, 26.4%, 25.2%, 42.6%) for the existing global dataset. +If we further double the statistics of the Daya Bay Asimov +data–as expected from the full Daya Bay dataset—in conjunc- +tion with the modified binning in fission fractions, we find + +7 +0 +1 +2 +3 +4 +5 +6 +7 +8 +/fission] +2 +cm +-43 + [10 +239 +σ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: -0.990 +6 +6.2 +6.4 +6.6 +6.8 +/fission] +2 +cm +-43 + [10 +235 +σ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: 0.013 +0 +2 +4 +6 +8 +10 +12 +14 +/fission] +2 +cm +-43 + [10 +238 +σ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: 0.757 +FIG. 3. Isotopic IBD yield fits for the existing global dataset with loose (75%) external constraints on the 241Pu IBD yield, σ241. Contours +are pictured for σ241 relative to the other isotopic yields, with the fit marginalized over the non-pictured isotopes. Correlation coefficients +between fitted σ241 and the other yields are given in the plot legends. Also shown in dashed lines are the theoretical IBD yields predicted by +the Huber-Mueller model. Stars indicate IBD yields chosen for illustration in Fig. 4. +5.65 +5.7 +5.75 +5.8 +5.85 +5.9 +5.95 +6 +6.05 +6.1 +/fission] +2 + cm +-43 +IBD yield [10 +Nominal IBD yields +Modified IBD yields +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +235 +F +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +239 +/F +241 +F +Daya Bay +RENO +FIG. 4. Top: IBD yield sets for two hypothetical LEU measure- +ments: one assuming measurements align with isotopic IBD yields +matching the best-fit for the existing global dataset, and another as- +suming alignment with σ239 and σ241 values matching those indi- +cated in Figure 3. The latter scenario’s values lie outside of the 1 +σ region preferred by the global IBD yield dataset; for this scenario, +σ238 is reduced to enable better vertical alignment of the two datasets +and easier comparison of slopes. Bottom: Ratio (F241/F239) of the +fission yields of 241Pu and239Pu for the hypothetical LEU dataset. +Realized F235 ranges for RENO and Daya Bay datasets are also pic- +tured. +a further improvement in precision to (1.3%, 21.7%, 16.4%, +30.8%). Thus, we conclude that it may be worthwhile for +Daya Bay to consider a more diversified fuel content binning +scheme in a future analysis of its final full-statistics IBD yield +dataset. This observation may also be applicable to other high- +statistics datasets spanning many LEU reactor cycles, such as +those recorded by RENO and DANSS [12]. +IV. +FUTURE IMPROVEMENTS FROM NEW +MEASUREMENTS AT MULTIPLE CORE TYPES +We now turn to consideration of future improvements in +global knowledge of isotopic IBD yields by performing new +measurements at a range of different reactor core types. We +will begin by considering the most imminently-achievable +next steps: short baseline measurements of a single LEU +core over a full fuel cycle, and a subsequent systematically- +correlated measurement at an HEU using the same νe de- +tector. +We will then proceed to study possible improve- +ments gained by making measurements at mixed-oxide and +plutonium-burning fast reactor core types. +A. +Benefits of New HEU and LEU Measurements +Some benefits of new measurements of IBD yields at short +distances from a full LEU reactor core cycle have already been +discussed in the literature [61], and have served as part of the +physics motivation for the NEOS-II experiment [89]. In par- +ticular, this configuration enables access to a wider range of +F239 and F235 values beyond those achieved at θ13 exper- +iments sampling multiple cores, which should result in im- +proved σ239 constraints. When coupled with a systematically- +correlated HEU-based measurement, which could be achieved + +8 +via two site deployments of the same detector system, di- +rect constraints on σ238 may exceed the claimed precision +of the summation prediction of Mueller et al. [59]. Multi- +ple current or near-future efforts, such as PROSPECT-II [79] +or MAD [78], are well-suited to realize part or all of this com- +bined LEU-HEU measurement program. +Such a setup would also broaden access to LEU fuel content +regimes with less linear relationships between F239 and F241, +allowing for improved constraint of σ241. This improvement +was demonstrated above for the hypothetical LEU measure- +ments in Figure 4. Realized effective F239 ranges for Daya +Bay and RENO are also highlighted with shaded bands; we +note that offsets in median F235 (and, while not pictured, also +F241/F239) between hypothetical LEU and Daya Bay/RENO +cases is due to the specifics of the single cycle core loading +simulated in Ref. [71]. A new short-baseline LEU measure- +ment set can capture periods earlier and later in the fuel cycle +of a conventional LEU core with respect to RENO and Daya +Bay, when relative contributions of 239Pu and +241Pu fissions +deviate most strongly from their cycle-integrated mean. For +the hypothetical short-baseline LEU measurement, F239/F241 +varies roughly 6%, from 17% to 23%, over a cycle. Daya +Bay’s and RENO’s F241/F239 ratios, meanwhile vary by only +3% or less, with maximums and minimums of 20% and 17%, +respectively [25, 55]. +The extent to which these HEU and LEU measurements can +improve constraints on σ241 has so far not been investigated in +the literature. To do so, we apply the four-parameter yield fit +of Eq. 5 to the hypothetical HEU and LEU datasets described +in Section II B, Table I, and Figure 1. Table II gives the result- +ing precision in measurements of the four isotopic IBD yields +probed by this new HEU+LEU dataset. The most striking dif- +ference with respect to the current global dataset is the sub- +stantial improvement in knowledge of 239Pu and 241Pu yields. +Uncertainties in σ239 and σ241 are improved from 25.2% and +42.6% in the existing dataset to 4.6% and 10.5%, respectively, +greater than four-fold improvement in both values. As illus- +trated in Figure 5, this improvement can be partially attributed +to the reduction in degeneracy between these two isotopes’ fis- +sion fraction variations over a full LEU fuel cycle. If all mea- +surements are instead performed with a 1 ton detector, more +closely approximating the expected size of the MAD detec- +tor, uncertainties are similar in size, with σ235,238,239,241 shift- +ing from (1.6%, 11.2%, 4.6%, 10.5%) for the PROSPECT-II +sized detector case to (1.62%, 11.7%, 6.1%, 14.6%) for the +MAD detector case. Thus, the HEU+LEU deployment sce- +nario may yield major benefits for both physics-oriented or +smaller applications-oriented future detectors. +As noted in Ref. [61], σ238 constraints are also significantly +improved, primarily due to the correlated nature of the detec- +tor systematics assumed between the HEU and LEU measure- +ments. If this correlation is removed, or if the chosen opti- +mistic 1% HEU thermal power uncertainties are increased to +the currently-achievable 2% level, precision in knowledge of +the 238U yield is substantially reduced – to 18.1% and 17.2% +for these two cases, respectively – while precision in knowl- +edge of the 241Pu yield is virtually unchanged. Thus, follow- +ing the next generation of short-baseline HEU and LEU mea- +surements, the precision of knowledge of the 241Pu yield may +rival that of its sub-dominant 238U counterpart, and will be +less dependent on a detailed understanding of host reactors’ +thermal powers and on movement-induced changes in detec- +tor response. At this point, direct νe -based measurements of +241Pu fission attributes may begin to have useful application +in testing the general accuracy of nuclear data knowledge for +this isotope – similar to the value provided by νe -based con- +straints of 238U from the current global dataset. +B. +Benefits from MOX Reactor Measurements +Reactors burning mixed-oxide (MOX) fuels are another +promising venue for performing IBD yield measurements +with unique Fi combinations. +In particular, the RG-MOX +measurement case may be an imminently realizable one, given +the presence and operation of RG-MOX commercial cores in +Europe and Japan. The 50% reactor-grade mixed-oxide (RG- +MOX) core described in Section II B features F239 far higher +than an LEU core and broad variations in F241 from nearly +15% at reactor start-up to roughly 25% after one cycle. Ra- +tios F239/F241 vary much more widely from cycle beginning +(27%) to end (45%) compared to the LEU reactor case above. +Amidst these substantial fission fraction variations, 238U frac- +tions remain relatively consistent between LEU and RG-MOX +cases, offering further opportunity for reduction in degeneracy +between 238U and the other isotopes. +Addition of a hypothetical ten-datapoint IBD yield dataset +from this RG-MOX reactor core provides substantial enhance- +ments in IBD yield precision when added to those of the short- +baseline HEU and LEU datasets, which are also summarized +in Table II. Expected precision of yields σ239 and σ241 are im- +proved by another factor of ∼ 2 and ∼ 3 respectively when the +hypothetical RG-MOX is added to the fit alongside the hypo- +thetical HEU and LEU datasets. Meanwhile, σ238 yield preci- +sion is also tightened to 9.7% expected relative uncertainty. +Correlations between yield fit parameters for this case are +also pictured in Figure 5, and appear further reduced between +239Pu and 241Pu with respect to the hypothetical HEU+LEU +case. As with the HEU+LEU case, if measurements are per- +formed instead with a MAD-sized 1 ton detector target, only +modest degradation in precision is seen: σ235,238,239,241 un- +certainties shift from (1.6%, 9.7%, 2.2%, 3.4%) for a 4 ton +target to (1.6%, 10.3%, 2.5%, 3.9%) for a 1 ton target. un- +certainty. On the other hand, if the correlation between the +reactor measurements are removed, or if the chosen opti- +mistic 1% HEU thermal power uncertainties are increased to +the currently-achievable 2% level, precision in knowledge of +the 238U and 241Pu yields are reduced—to 14.9%, 15.4% and +4.3%, 5.0% respectively— and are moderately worse than the +theoretical yields. Comparing this with the HEU+LEU case +where the precision achievable on 238U yield is 11.1%, the +improvement provided by the addition of RG-MOX reactor +data doesn’t fully compensate for the loss in precision due to +the lack of correlation or a reduction in thermal power uncer- +tainty. +With measurements at three reactor types – HEU, LEU, and + +9 +Case +Description +Precision on σi (%) +235U 238U 239Pu 240Pu 241Pu +- +Existing Global Data +1.3 +26.4 +25.2 +- +42.6 +1 +HEU + LEU +1.6 +11.1 +4.6 +- +10.5 +3 +HEU + LEU + RG-MOX +1.6 +9.7 +2.2 +- +3.4 +2 +HEU + LEU + WG-MOX +1.6 +9.9 +2.5 +- +3.6 +4 +HEU + LEU + Fast +1.6 +10.9 +4.6 +27.2 +10.3 +5 +All +1.6 +9.5 +2.1 +23.6 +3.3 +6 +All, Uncorrelated +1.5 +14.3 +2.1 +36.2 +4.2 +- +Model Uncertainty [66] +2.1 +8.2 +2.5 +- +2.2 +TABLE II. Constraints on IBD yields of 235U, 238U, 239Pu, 240Pu, and 241Pu, from future hypothetical datasets from LEU and HEU reactors, +given as a percentage of the best fit yield. For all cases unless noted, detector systematic uncertainties are assumed to be correlated between +measurements, and a 75% external constraint is used for 241Pu and for 240Pu when applicable. The ‘All’ case considers inclusion of HEU, LEU, +RG-MOX, VTR and PFBR yield measurements employing the same detector. Model prediction uncertainties from [66] are also provided. +5.8 +6 +6.2 +6.4 +/fission] +2 +cm +-43 + [10 +235 +σ +7 +8 +9 +10 +11 +12 +13 +14 +/fission] +2 +cm +-43 + [10 +238 +σ +Correlation: -0.383 +5.8 +6 +6.2 +6.4 +/fission] +2 +cm +-43 + [10 +235 +σ +4 +4.1 +4.2 +4.3 +4.4 +4.5 +4.6 +4.7 +4.8 +/fission] +2 +cm +-43 + [10 +239 +σ +Correlation: 0.772 +5.8 +6 +6.2 +6.4 +/fission] +2 +cm +-43 + [10 +235 +σ +5.2 +5.4 +5.6 +5.8 +6 +6.2 +6.4 +6.6 +6.8 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: 0.727 +7 +8 +9 +10 +11 +12 +13 +14 +/fission] +2 +cm +-43 + [10 +238 +σ +4 +4.1 +4.2 +4.3 +4.4 +4.5 +4.6 +4.7 +4.8 +/fission] +2 +cm +-43 + [10 +239 +σ +Correlation: -0.511 +7 +8 +9 +10 +11 +12 +13 +14 +/fission] +2 +cm +-43 + [10 +238 +σ +5.2 +5.4 +5.6 +5.8 +6 +6.2 +6.4 +6.6 +6.8 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: -0.681 +4 +4.1 4.2 +4.3 +4.4 +4.5 +4.6 +4.7 +4.8 +/fission] +2 +cm +-43 + [10 +239 +σ +5.2 +5.4 +5.6 +5.8 +6 +6.2 +6.4 +6.6 +6.8 +/fission] +2 +cm +-43 + [10 +241 +σ +Correlation: 0.490 +FIG. 5. Isotopic IBD yield contours for a combined fit of hypothetical HEU, LEU, and RG-MOX datasets. In each panel, fits are marginalized +over the undepicted isotopes. Correlation coefficients between each pair of isotopes are provided in the legend. +MOX – with a common detector, direct IBD-based constraints +on νe production by the four primary fission isotopes may +be expected to rival or exceed the precision of conversion- +based predictions. Most of these direct isotopic yield uncer- +tainties are also smaller and more well-defined in origin than +the O(5%) uncertainty attributed to summation predictions for +these isotopes. Thus, with a global HEU+LEU+MOX dataset, +one could generate IBD-based reactor νe flux predictions for +many existing or future reactor types free from biases known +to be present in conversion-predicted models without sacrific- +ing relative model precision. +Expected isotopic IBD yield measurement precision de- +livered by instead combining a ten datapoint weapons-grade +mixed-oxide (WG-MOX) measurement with the hypothetical +HEU and LEU datasets has also been considered. IBD yield +uncertainties for a HEU+LEU+WG-MOX measurement set +are slightly worse than a HEU+LEU+RG-MOX set for σ238, +σ239, and σ241 as shown in Table II. Similarity in results be- +tween MOX fuel types should not be too surprising, since both +WG-MOX and RG-MOX cycles roughly span a ∼ 16−17% + +10 +range in F239/F241 fission fraction ratios. +It is worth noting that wide variations in F239/F241 should +also expected to be provided by conventional LEU cores burn- +ing entirely fresh fuel, such as would occur upon first oper- +ation of a new commercial power plant [90]. In this case, +F239/F241 fission fraction ratios should be expected to vary +by well over 10% over course of a fuel cycle [76]. Thus, in +lieu of MOX-based options, IBD yield measurement regimes +including newly started commercial cores likely serve as an- +other promising avenue for producing precise constraints on +all main fission isotopes. +C. +Benefits from Fast Reactor Measurements +Since fast fission cross-sections of many minor actinides +– particularly 240Pu – are substantially higher than ther- +mal fission cross-sections, fission fractions in the VTR and +PFBR fast reactors are substantially different than those of +the high-MOX-fraction conventional core configurations de- +scribed in [72]. In particular, 240Pu fissions now compose a +non-negligible fraction of the total, and, as a result, 241Pu fis- +sion fractions are substantially lower. The addition of the two +fast reactor dataset to the hypothetical HEU and LEU datasets +is also summarized in Table II. The most striking product of +introducing these datasets to the fit is the potential for set- +ting the first-ever meaningful constraints on νe production +by 240Pu. We find roughly comparable 240Pu yield measure- +ments when either VTR or PFBR are fitted separately with the +other datasets. Such a measurement could prompt new and +deeper study of fission yields and decay data for this minor +actinide, which plays a major role in the operation of next- +generation fast reactor systems. The level of achievable preci- +sion in the σ240 measurement is primarily driven by the preci- +sion in understanding the thermal output of these fast reactor +cores – an instrumentation challenge under active investiga- +tion in the nuclear engineering community. +Inclusion of fast reactor datasets generates only minor im- +provements in the knowledge of σi for the other primary +fission isotopes beyond that achievable with the HEU+LEU +measurement scenario. While this results primarily from the +general lack of knowledge of the value of σ240, it also high- +lights the value delivered by multiple highly systematically +correlated measurements at differing fuel composition, like +that provided by the MOX reactor cases, in contrast to the sin- +gle measurement provided by the relatively static composition +of these fast reactor cores. Were F240 to evolve in a meaning- +ful way for either core, it is likely that the isotopic IBD yield +knowledge delivered by this core would be substantially im- +proved. +V. +DISCUSSION AND SUMMARY +After observing that the current global IBD yield dataset +exhibits some capability to constrain antineutrino production +by 235U, 238U, 239Pu, and 241Pu, we have investigated how +suites of future systematically-correlated measurements at di- +verse reactor core types can improve knowledge for these +and other fission isotopes. We have observed that with the +simplest combination of correlated HEU and LEU measure- +ments using a PROSPECT-sized or MAD-sized IBD detec- +tor, an IBD yield measurement precision of 12% or better can +be achieved for all four fission isotopes. With a combina- +tion of HEU, LEU, and RG-MOX datasets, all isotopic yields +can be directly measured with a precision rivaling or exceed- +ing the precision claimed by conversion-predicted models. If +measurements of fast reactors are also included in the global +dataset, first constraints of order 25% precision can be placed +on antineutrino production by 240Pu. Beyond future measure- +ments, we also noted other avenues for improving knowledge +of isotopic IBD yields with current data: in particular, mea- +surements performed over multiple LEU fuel cycles, such as +Daya Bay and DANSS, can benefit from exploiting known +variations in 241Pu between cycles. +With a combined global dataset in hand from multiple +reactor types, one can generate IBD-based reactor νe flux +predictions for many existing or future reactor types free +from biases known to be present in conversion-predicted +models without sacrificing in relative model precision. +If +one considers the full suite of correlated HEU, LEU, RG- +MOX and fast reactor measurements (the “All” scenario +in Table II), the resultant data-based model would include +(σ235, σ238, σ239, σ240, σ241,) uncertainties of (1.6, 9.5, 2.1, +23.6, 3.3)%. +The correlation between these achievable +directly-constrained uncertainties has also been calculated, +and can be seen in Figure 6, alongside those of the Huber- +Mueller model [91]. Besides representing the similar mag- +nitudes in uncertainty, Figure 6 shows direct measurements’ +reduced correlations between 235U, 239Pu, and 241Puwith re- +spect to conversion predictions, which are primarily caused +by the common experimental apparatus used at ILL for input +fission beta measurements [92, 93]. +This kind of direct and precise understanding of all of the +major fission isotopes’ contributions to reactor antineutrino +emissions would represent movement into an era of ‘preci- +sion flux physics’ offering many potential pure and applied +physics benefits. On the applications side, it would enable +unbiased, high-fidelity monitoring, and performing of robust +case studies for, a broad array of current and future reactor +types. Well-measured isotopic antineutrino fluxes could be +compared to summation-predicted ones to provide enhanced +benchmarking and improvement of nuclear data associated +with the main fission isotopes and their daughters, as well +as the first meaningful integral datasets for validating the nu- +clear data of 240Pu. +These models and correlated datasets +would allow for precise independent tests of each of the four +IBD yield predictions provided by the Huber-Mueller model, +enabling thorough investigation of the hypothesis that mis- +modelling of one or more isotopes’ yields is responsible for +the reactor antineutrino anomaly. Precise and reliable IBD- +based flux constraints would also improve the reach of be- +yond standard model searches with signal-dominated coherent +neutrino-nucleus scattering detectors [3]. Finally, by probing +for persistent residual IBD yield deficits common to all iso- + +11 +1.6 +-2.4 +1.6 +3.6 +2.0 +-2.4 +9.5 +-3.2 +-12.8 +-4.6 +1.6 +-3.2 +2.1 +3.6 +1.9 +3.6 +-12.8 +3.6 +23.6 +7.0 +2.0 +-4.6 +1.9 +7.0 +3.3 + +235 +U + +238 +U + +239 +Pu + +240 +Pu + +241 +Pu + + +235 +U + +238 +U + +239 +Pu + +240 +Pu + +241 +Pu + +15 +− +10 +− +5 +− +0 +5 +10 +15 +20 +25 +Uncertainty [%] +2.4 +2.6 +2.5 +8.2 +2.6 +2.9 +2.7 +100.0 +2.5 +2.7 +2.6 + +235 +U + +238 +U + +239 +Pu + +240 +Pu + +241 +Pu + + +235 +U + +238 +U + +239 +Pu + +240 +Pu + +241 +Pu + +15 +− +10 +− +5 +− +0 +5 +10 +15 +20 +25 +Uncertainty [%] +FIG. 6. Left: Uncertainties in isotopic IBD yield measurements based on a hypothetical global dataset including HEU, LEU, RG-MOX, and +fast reactor IBD yield measurements. Diagonal elements correspond to the uncertainty in isotopic yields given for the “All” case in Table II, +while off-diagonal elements describe the correlations between them. The values are extracted by taking the square root of the corresponding +elements of the correlation matrix and are assigned a negative value where the correlations are negative. Full covariance matrices are provided +in the supplementary materials accompanying this paper. Right: Uncertainties in IBD yields predicted by the Huber-Mueller model [66]. Since +there are no theoretical models predicting σ240, we assign 100% uncertainty on it. +topes with respect to conversion or summation models, the +community can search for enduring hints of sterile neutrino +oscillations, even in the presence of other confounding effects, +such as neutrino decay or wave packet de-coherence [94]. 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Soc. 77, 1118 (2020), +arXiv:2009.13355 [physics.ins-det]. +[90] A schedule of forecasted reactor facility start-ups world- +wide +are +given +at +https://world-nuclear.org/information- +library/current-and-future-generation/plans-for-new-reactors- +worldwide.aspx. +[91] Both hypothetical future and Huber-Mueller model IBD yield +covariance matrices are included in supplementary materials +accompanying this paper. +[92] F. Von Feilitzsch, A. A. Hahn, +and K. Schreckenbach, +Phys.Lett. B118, 162 (1982). +[93] A. A. Hahn et al., Phys.Lett. B218, 365 (1989). +[94] C. A. Arg¨uelles, I. Esteban, M. Hostert, K. J. Kelly, J. Kopp, +P. A. N. Machado, I. Martinez-Soler, and Y. F. Perez-Gonzalez, +(2021), arXiv:2111.10359 [hep-ph]. + diff --git a/NdFPT4oBgHgl3EQflzWN/content/tmp_files/load_file.txt b/NdFPT4oBgHgl3EQflzWN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17e9b7844f6b8c4dd85dc10b5651b49f1af8bfd3 --- /dev/null +++ b/NdFPT4oBgHgl3EQflzWN/content/tmp_files/load_file.txt @@ -0,0 +1,1240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf,len=1239 +page_content='Exploring Current Constraints on Antineutrino Production by 241Pu and Paths Towards the Precision Reactor Flux Era Yoshi Fujikake, Bryce Littlejohn,* and Ohana B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Rodrigues† Physics Department, Illinois Institute of Technology, Chicago, IL 60616, USA Pranava Teja Surukuchi‡ Wright Laboratory, Yale University, New Haven, CT 06520, USA By performing global fits to inverse beta decay (IBD) yield measurements from existing neutrino experi- ments based at highly 235U enriched reactor cores and conventional low-enriched cores, we explore current direct bounds on neutrino production by the sub-dominant fission isotope 241Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For this nuclide, we determine an IBD yield of σ241 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='16 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='47 cm2/fission, a value (135 ± 58)% that of current beta conversion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This constraint is shown to derive from the non-linear relationship between burn-in of 241Pu and 239Pu in con- ventional reactor fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By considering new hypothetical neutrino measurements at high-enriched, low-enriched, mixed-oxide, and fast reactor facilities, we investigate the feasible limits of future knowledge of IBD yields for 235U, 238U, 239Pu, 241Pu, and 240Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We find that first direct measurement of the 240Pu IBD yield can be performed at plutonium-burning fast reactors, while a suite of correlated measurements at multiple reactor types can achieve a precision in direct 238U, 239Pu, and 241Pu yield knowledge that meets or exceeds that of current theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' INTRODUCTION A nuclear reactor primarily generates thermal energy as product nuclei inherit (as kinetic energy) and deposit (through repeated elastic collisions) excess rest mass energy from the fission of heavy nuclides in the reactor’s fuel, such as 235U, 238U, 239Pu, 241Pu, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Successive decays of these neutron-rich product nuclei release additional energy in the form of beta particles, gamma-rays, and antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' While the two former product types are additional, sub-dominant contributors to heat generation in a reactor, the antineutri- nos (νe ) and their associated kinetic energy entirely escape the reactor core, offering an attractive avenue for studying the properties of neutrinos [1–3], interrogating state-of-the- art nuclear data [4], and non-intrusively monitoring nuclear reactor cores [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Reactor-based νe detectors have demon- strated that neutrinos have mass [6–9], and have searched for the existence of new heavy neutrino states [10–15] and other new physics phenomena [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By observing discrepan- cies with respect to existing theoretical νe flux and energy spectrum predictions, they have also highlighted limitations of and/or inaccuracies in community fission yield and beta decay databases [7, 9, 24–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Antineutrino monitoring case studies have explored a variety of potential use case scenarios, such as thermal power load-following and determination of reactor fissile inventory [33–37], and existing νe detectors have con- firmed the feasibility of some of these activities [25, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The average number of antineutrinos released or detected per nuclear fission depends on the fission isotope in question: different fission isotopes have different fission product yields, with each product varying in its distance from the line of sta- bility and having its own unique nuclear structure and decay blittlej@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='edu † obenevidesrodrigues@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='edu ‡ pranavateja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='surukuchi@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='edu scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, reactor cores with differing fuel compositions are expected to differ in their rate of νe output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' These ex- pected differences have been explicitly demonstrated in recent νe experiments using the inverse beta decay (IBD) interaction process, p + νe → e++ n, which has a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 MeV interaction threshold and a precisely-predicted cross-section, σIBD(Eν), versus νe energy Eν [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For these experiments, measured νe fluxes have been expressed in terms of an IBD yield per fission σf [1]: σf(t) = � i Fi(t)σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' (1) In this expression, Fi(t) is the fraction of fissions contributed by isotope i in the sampled reactor core(s) during the experi- ment’s measurement period and σi its IBD yield per fission, σi = � Si(Eν)σIBD(Eν)dEν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' (2) Here, Si(Eν) is the true produced νe energy (Eν) spectrum per fission for isotope i, and σIBD is the inverse beta decay interaction cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In a straightforward demonstration of variations in νe emis- sion between fission isotopes, reported IBD yields σf are clearly offset [41] between measurements at 235U-burning highly enriched reactor cores [42–48] and measurements per- formed at commercial cores burning a mixture of 235U, 238U, 239Pu, and 241Pu [7, 49–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In a separate demonstration, the Daya Bay and RENO experiments have compared IBD yields measured in the same detectors at differing points in their sam- pled commercial reactors’ fuel cycles, observing higher yields during periods with higher (lower) 235U (239Pu) fission frac- tions [25, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By performing fits to a set of σf measurements at reactors of well-known fission fraction Fi, one can use Equations 1 and 2 to directly determine the isotopic IBD yield σi of one or more fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' With a single HEU-based experi- ment, the IBD yield for 235U, σ235, can be trivially deter- mined as σ235 = σf, since F235 approaches unity for these arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='13123v1 [hep-ph] 30 Jan 2023 2 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' On their own, HEU-based σ235 measurements exhibit deficits [57] with respect to commonly-used beta-conversion predictions [58, 59], indicating issues in modeling either the core’s νe emissions or νe behavior during propagation [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Daya Bay and RENO σf measurements, which encompass multiple data points with differing LEU fuel composition F235, 238, 239, 241, when combined with modest theoretical constraints on σ238, 241, yields from the sub-dominant iso- topes 238U and 241Pu, enable determination of isotopic yields for both 235U and 239Pu [25, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' These measurements show a deficit with respect to 235U conversion predictions, but no such deficit for 239Pu, providing further credence to the νe emission mis-modelling hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Going further, global fits of both LEU and HEU datasets can be used to simultaneously determine σ235, 238, 239 [61]: the measured σ238 shows a sig- nificant (33±14)% deficit [62] with respect to summation pre- dictions based on community-standard nuclear databases [59], suggesting potential issues in current 238U fission yield mea- surements or evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Future direct determination of iso- topic IBD yields for a wider array of fission isotopes beyond 235U, 239Pu, and 238U, as well as improved precision for these three isotopes, can lead to further understanding or improve- ment of existing nuclear data, reactor νe models, and reactor- based fundamental physics studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Improved isotopic IBD yield measurements also hold po- tential benefits for future νe -based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Some ad- vanced reactor technologies present unique safeguards chal- lenges that may be satisfied by near-field νe -based monitor- ing capabilities [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' However, neutrino emissions have never been measured at advanced reactor cores, some of which dif- fer substantially from measured HEU and LEU reactor types in both fuel composition and core neutronics [35, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For example, mixed-oxide reactor fuels, which, unlike con- ventional low-enriched fuel, are produced from a mixture of uranium and plutonium isotopes, may be deployed in future reactors to realize a closed nuclear fuel cycle or as a means of disposing of existing plutonium stockpiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Fast fission reac- tor technologies, which, unlike conventional thermal reactors, rely on fast neutron induced fission to maintain criticality, may offer safety and sustainability advantages with respect to con- ventional reactor types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For these reactors, better direct deter- minations of true underlying σi can enable more robust and reliable future monitoring capabilities than would be possible using existing demonstrably imperfect models of νe produc- tion per fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In this paper, we study how existing and potential future IBD measurements can provide first direct glimpses at νe production by previously unexplored fission isotopes and im- prove our precision in understanding of the more-studied iso- topes 235U, 239Pu, and 238U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By performing loosely con- strained fits of isotopic IBD yields to existing LEU and HEU datasets, we demonstrate the feasibility of achieving non- trivial future bounds on νe production by 241Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By applying the same fit techniques to hypothetical future high-precision IBD yield measurements at HEU, LEU, MOX, and fast fis- sion reactors, we show that direct IBD yield determinations for all four primary fission isotopes (235U, 238U, 239Pu, and 241Pu) can meet or exceed the claimed precision of exist- ing conversion-based predictions while also placing the first meaningful bounds on 240Pu νe production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We begin in Section II with a description of the global fit and existing and hypothetical future IBD yield datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Re- sults of the fit to existing datasets and studies of 241Pu limits are presented and discussed in Sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In Section IV, we describe the set of considered future hypothetical experiments and the result of applying global fits to the hypothetical re- sults of these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Main results are then summarized in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' GLOBAL DATASETS AND FIT TECHNIQUE In this analysis we perform fits to a set of IBD rate measure- ments with varying degrees of systematic correlation between each measurement set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For an individual measurement, the number of IBD interactions N detected per time interval t can be described as: N = NpεP(L) 4πL2 � Wth(t)σf(t) ¯E(t) dt, (3) where Np is the number of target protons, ε is the efficiency of detecting IBDs, P(L) is the survival probability due to neutrino oscillations, and L is the core-detector distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Of the time-dependent quantities, Wth(t) is the reactor’s thermal power, ¯E(t) = � i Fi(t)ei is the core’s average energy re- leased per fission, ei is the average energy released per fission of isotope i, and Fi(t) and σf(t), as in Equations 1 and 2, are the fission yields and IBD yields of isotope i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In order to per- form one or multiple measurements of σf, a reactor νe flux experiment must measure N while characterizing the other reactor and detector inputs in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Existing Datasets Many experiments have successfully measured σf values and associated statistical and systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As in- put for this study, we include time-integrated IBD yield mea- surements and uncertainties reported by the Goesgen, Bugey- 3, Bugey-4, Rovno, Palo Verde, CHOOZ, and Double Chooz LEU-based experiments and the ILL, Savannah River, Kras- noyarsk, Nucifer and STEREO HEU-based experiments, as well as the highly-correlated datasets at varying Fi from the Daya Bay and RENO experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Calculated fission frac- tions and measured yields for these experiments, as well as as- sociated uncertainties and cross-measurement systematic cor- relations, have been summarized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [66], and are used for portions of this paper’s analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Input data tables are pro- vided in the public GitHub repository [67] provided by the authors as an accompaniment to this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Since we do not consider short-baseline oscillations as part of this analysis, reactor-detector baselines are not used in analysis of existing datasets, but are nonetheless provided in these tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Hypothetical Future Datasets For this study, we also generate hypothetical future IBD yield datasets and uncertainty budgets matching the expected capabilities of experimental deployments at HEU, LEU, MOX and fast reactor types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' These are also provided in the GitHub supplementary materials, along with assumed uncertainty co- variance matrices for all considered hypothetical measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Hypothetical IBD yield measurements are Asimov datasets free of statistical and systematic fluctuations that are generated according to Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As input to this equa- tion, fission fractions are required for each host reactor and are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To match general indications from recent summations [32] and fission beta [68], and νe flux evolution [25] measurements, and matching the approach in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [61], we choose input ‘true’ IBD yield values match- ing a scenario where Huber-Mueller modelled yields [69] are only incorrect for 235U: (σ235, σ238, σ239, σ241) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='05,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='10,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='40,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='03) ×10−43 cm2/fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The yield for 240Pu has not been predicted in the literature to our knowl- edge, so we estimate it by applying a 3Z-A scaling suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [70] to the four previously-mentioned isotopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' the determined central value is σ240 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='96 ×10−43 cm2/fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Other experimental assumptions regarding detector, reactor, and experimental layout parameters are then required to de- fine the statistical and systematic uncertainties associated with each hypothetical IBD yield measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The HEU-based measurement is modeled after the HFIR facility at Oak Ridge National Laboratory, sporting 85 MW of thermal power, a 100% 235U fission fraction, and a 7 m reactor-detector center-to-center distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' LEU-based mea- surements are assumed to occur at a 20 m center-to-center dis- tance from a core following the attributes of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 GWth Daya Bay core with an 18 month fuel cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Assumed fission frac- tions are chosen to fall roughly in the middle of the range reported for Daya Bay’s cores in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 1 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [75], and cor- respond to a fully-loaded core with roughly 1/3 of its rods con- taining fresh (pure uranium oxide at start-up) fuel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' this level of partial reloading is customary when operating cores of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' MOX-based measurements are modelled after the MOX reactor studies of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [72], and is assumed to occur 20 m from a core with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 GWth thermal power and 18 month cy- cle length, and fission fractions matching those of the sim- ulated 50% weapons-grade MOX-burning core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Weapons- grade (WG) plutonium is characterized by low 240Pu and 241Pu isotopic fractions, and thus a low F241 fission fraction at reactor start-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' These WG-MOX core parameters corre- spond to a realizable operational scenario implemented for the goal of plutonium stockpile disposition in a commercial reactor core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We will also reference a similar case where 50% reactor-grade (RG) MOX fuel is used in the same reactor type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' these parameters correspond to an operational scenario for a commercial complex operated as part of a closed nuclear fuel cycle program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Following recommendations of the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [72], fission fractions for the WG-MOX core exam- ple are assumed to match the reported fission fractions for the first third of pictured 50% WG-MOX running in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [72], while fractions from the RG-MOX case are assumed to match the those of 50% WG-MOX running between days 800 and 1350 [76];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' fission fractions were extracted by interpolating fission rates from this reference and normalizing such that the sum for the four primary fission isotopes is equal to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' It should be stressed that modeled fuel content evolution for LEU and MOX cores is highly dependent on the initial condi- tions of the fuel, on the neutronics of the involved core type, and on reactor operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In this study, we include one spe- cific fission fraction set for each fuel type – LEU, WG-MOX, and RG-MOX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' the impact or potential benefits of further vari- ations between LEU or MOX core types is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Finally, two experiments are assumed to occur at the base- lines of 20 m and 7 m distances from the primarily plutonium- burning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='25 GWth PFBR fast breeder reactor in India [73] and the 300 GWth Versatile Test Reactor fast reactor [74] re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The former reactor plays a central role in plans for realization of an independent, sustainable nuclear fuel cycle in India, while the latter has been developed as a US-based reactor materials and irradiation R&D facility based at Idaho National Laboratory [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Assumed reactor and site parame- ters for all measurements are summarized in Table I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' fission fraction values for all hypothetical measurement data points used in this study are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 0 5 10 15 20 25 30 Data Points 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 1 Fission Fraction Fission fractions in PROS HEU+LEU+WGMOX+RGMOX+VTRRx+PFBR U235 U238 Pu239 Pu240 Pu241 HEU LEU WG MOX RG MOX VTR PFBR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Fission fractions used for hypothetical future measurement data points described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For all experiments, an IBD detector matching qualities of the 4 ton PROSPECT reactor νe detector are used [77];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' rel- evant parameters are also listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In some cases, a 1 ton detector with otherwise similar experimental parame- ters is also considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' this case enables investigation of the value of using a near-future compact νe monitoring detector, such as the Mobile Antineutrino Demonstrator [78] (MAD), to perform IBD yield benchmarking measurements at multi- ple reactor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In all cases, the statistical uncertain- ties associated with each datapoint for each reactor-detector combination are estimated using the associated detector and reactor parameters quoted in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' I and lie between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='15 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For simplicity, we do not consider statistical 4 Parameter HEU LEU MOX Fast (PFBR) Fast (VTR) Reactor Thermal Power (MWth) 85 2900 3200 1250 300 Burnup Profile [71] [72] [73] [74] Reactor Cycle Length 24 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 y 100 d Experimental Core-Detector Distance (m) 7 m 20 m 20 m 20 m 20 m Data-Taking Length 3 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 y 1 y 100 d Detector Active Mass 4 ton (1 ton) Target Protons 2×1029 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5×1029) IBD Detection Efficiency 40% Uncertainty, Reactor Thermal Power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0% Fission Fractions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% Energy per Fission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% Uncertainty, Detector Target Protons 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0% Detection Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='75% IBD Cross Section 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1% Total Reactor Systematic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% Total Detector Systematic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3% TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Assumed reactor and site parameters for the hypothetical future short-baseline reactor experiments described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' and systematic uncertainty contributions from IBD-like back- grounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' for a PROSPECT-like detector expecting signal-to- background ratios of better than 4 (10) deployed on-surface at an HEU (LEU) reactor [79], IBD counts would be expected to dominate measurement statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Hypothetical measurements should also be accompanied by predicted reactor- and detector-related systematic uncertain- ties, which are also summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Systematics for most cores are dominated by the uncertainty in the thermal power produced by the operating core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Commercial reactor companies have provided sub-percent precision in reported thermal powers for existing IBD yield measurements at nu- merous reactor sites [80, 81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' for this reason, we choose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5% uncertainty for LEU and MOX cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' While similar thermal power measurement devices and strategies could be applied to HEU facilities, in practice, legacy systems used in exist- ing HEU facilities have recently provided thermal power un- certainties closer to 2% [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' for this analysis, we optimisti- cally assume implementation of upgraded measurement sys- tems or techniques capable of providing 1% precision at an HEU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Advanced technologies for time-stable and high- precision thermal power monitoring in sodium-cooled fast re- actors like PFBR and VTR are under active development, due to the difficulties associated with the coolant’s high temper- ature and chemical corrosiveness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' given the lack of available quantitatively demonstrated capabilities, we also assume a 1% thermal power uncertainty for this core type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' While thermal power uncertainties for different reactors are assumed to be uncorrelated, this uncertainty is correlated between multiple measurements at the same core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Measured IBD yields for an experiment will also be uncertain due to the limits in knowl- edge of fission fractions in the core, which is defined via de- tailed reactor core simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In the absence of these cal- culations for all core types, we will assume an uncertianty of 0% for the HEU experiment and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% for all other cores, following the value quoted by Daya Bay and others for LEU cores [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This uncertainty is also assumed to be uncorre- lated between cores, but correlated between measurements at the same core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Isotopic energy release per fission ei – re- quired for calculating expected experiment statistics – have minor IBD yield uncertainty contributions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% depending on core fuel content [82];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' the ei central value and uncertainty for 240Pu is assumed to match that of 241Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' On the detector side, uncertainties are dominated by the limited knowledge of IBD detection efficiency, assumed to be known with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='75% precision, as well as knowledge of the to- tal number of protons within the detector’s target region, as- sumed to be known to 1%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' these chosen values reflect those achieved in a range of recent large- and small-detector IBD experiments [48, 55, 83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In this analysis, we consider the possibility of moving a single reactor neutrino detector to multiple reactor core types to perform systematically corre- lated IBD yield measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' for this reason, unless other- wise mentioned, we treat detector systematic uncertainties as correlated between all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Global Fit Approach To obtain isotopic IBD yields in this analysis, we use a least-squares test statistic: χ2 = � a,b � σf,a − r � i Fi,aσi � V−1 ab � σf,b − r � i Fi,bσi � + � j,k (σth j − σj)V−1 ext,jk(σth k − σk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' (4) In this fit, experimental inputs Fi and σf are as described above, and the sum i is run over five fission isotopes, 235U, 238U, 239Pu, 241Pu, and 240Pu, with five attendant IBD yield fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The experimental covariance matrix V de- fines the uncertainties for each experiment and their cross- correlations, as described in the previous-subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The final term is used to constrain fitted σi values to theoreti- cal predictions by adding a penalty that increases as the two quantities diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In contrast to most recent global IBD yield fits [25, 57, 85], we are interested in examining weakly- constrained or un-constrained simultaneous fits of all relevant fission isotopes’ IBD yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For this reason, j and k sum only over the three sub-dominant isotopes, 238U, 241Pu, and 240Pu, and the 3×3 V−1 ext is diagonal (no assumed uncertainty correla- tion between isotopes), with elements set to achieve wide 1σ theoretical constraints of 75% of the predicted yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To com- pare to previous IBD yield fits [61, 86], we occasionally con- sider the much tighter (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%) bounds on σ241 quoted by the Huber model [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For fits not involving fast reactor datasets, σ240 is pegged to the theoretically-predicted value, and has no effect on the subsequent 4-parameter fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' FITS TO EXISTING DATASETS AND 241PU IBD YIELD CONSTRAINTS We first consider IBD yield fits applied to the existing global yield datasets described briefly in Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' By first applying tight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% constraint on 241Pu, we largely reproduce unconstrained 235U, 238U, and 239Pu yield best-fit values re- ported for the oscillation-free fit in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Test statistic values with respect to the best-fit (∆χ2) versus input value are shown for each isotope in Figure 2, while minimizing over the three other isotopic yield parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We observe a best- fit 235U yield more than 3σ (5%) below the Huber-predicted value, and a best-fit 238U yield that deviates from the pre- dicted central value by (36±20), slightly more than in previ- ous fits [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As in previous fits, the 239Pu yield is found to be consistent with Huber-predicted values within a 5%, ∼ 1σ un- certainty band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This similarity in results indicates that the rel- atively new STEREO data point [48], while qualitatively bol- stering confidence in historical observations of a ∼5% yield deficit at HEU cores [87], has fairly modest quantitative im- pact on the primary issues surrounding data-model agreement for conversion-predicted uranium IBD yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' With consistency established with respect to previous anal- yses, we proceed with loosening of yield constraints for all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 i R 0 1 2 3 4 5 6 7 8 9 10 2 χ ∆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='013 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='953 235 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='201 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='642 238 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='044 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='047 239 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='026 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='001 241 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 i R 0 1 2 3 4 5 6 7 8 9 10 2 χ ∆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='013 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='953 235 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='264 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='730 238 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='252 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='910 239 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='426 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='353 241 σ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Isotopic IBD yield fit results for the existing global dataset with tight (top, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%) and loose (bottom, 75%) external constraints on the 241Pu yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Test statistic values with respect to the best-fit (∆χ2) are shown versus input value for each of the four primary fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For each isotope’s curve, the fit is marginalized over the other isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Figure 2 shows reported isotopic ∆χ2 test statistic values versus input σ value for each isotope while applying a looser constraint on 241Pu of 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Best-fit param- eters and 1σ ranges are found to be: σ235 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' σ238 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='37 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' σ239 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' σ241 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='16 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' (5) The best-fit χ2 min is found to be 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 for 38 degrees of free- dom (41 data points, 3 fit parameters), indicating an accept- 6 able goodness-of-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' However, this value is only slightly lower than that provided by the more-constrained fit (χ2 min = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6), indicating that this enhanced freedom has not sub- stantially improved data-model agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Central values of 235U, 238U, and 239Pu fit parameters are relatively sta- ble, remaining within 15% of those provided by the more- constrained fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Meanwhile, the newly freed 241Pu yield in- creases by 35%, although σ241 nonetheless remains consistent with its model-predicted value within its large 43% relative uncertainty band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus it appears that the current global IBD yield dataset does not have the statistical power to provide meaningful tests of underlying modelling issues for 241Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The disappointing lack of new insight should not be too sur- prising, given the small (O(5%) or less) fractional contribu- tion of 241Pu fissions in all existing measured reactor cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' However, it is interesting to note that σ241 1σ error bands are found to be tighter than the externally-applied constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This indicates that there are features in the existing global dataset that provide the power to specifically constrain 241Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To attempt to identify these features, we examined correla- tions between fitted isotopic yields, which are depicted in Fig- ure 3 as best-fit parameter space regions in two dimensions be- tween 241Pu and the other three isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Substantial 241Pu- 239Pu and 241Pu-238U degeneracies can be observed, with the former reflected in a more than five-fold increase in uncertain- ties on σ239 between the more-constrained (4% uncertainty) and less-constrained (25% uncertainty) fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Degeneracies can also be expressed by calculating correlation coefficients be- tween the fitted yield parameters, which are also given in the legends of Fig 3: ρσi,σj = (σi − σi)(σj − σi) σσiσσj (6) The extreme 241Pu-239Pu correlation can be understood by observing the fission fraction evolution trends experienced by LEU reactors, as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In these cores, F239 and F241 rise in tandem with reactor fuel burn-up, making it hard for unconstrained fits to simultaneously determine both σ239 and σ241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' It can also be understood as a simple reality of underlying nuclear physics in the core: 241Pu is produced by via multi-neutron capture on 239Pu, and thus its build-up in the core is dependent on the build-up of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In aspects of previous multi-datapoint LEU analyses, such as those of Daya Bay [25, 88] and RENO [55], 241Pu and 239Pu fission fractions are treated as explicitly linearly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We examine the limits of this linear correlation by gener- ating hypothetical LEU reactor IBD yield datasets following the method described in Section II B and fission yields from Figure 1: one dataset assumes isotopic yields matching the best-fit for the existing global dataset, and the other assumes true 241Pu and 239Pu yields close to the axis of anti-correlation between the two datasets, but beyond the 1σ bounds allowed by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Chosen true yields for this test are illustrated in the right panel of Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' the 238U yield for this case, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 cm2/fission, was chosen to vertically align the two yield datasets for easier comparison of trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Hypothetical yields for these two cases are pictured in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The test cases clearly differ in the change in slope, or curvature, present in the LEU data points, providing an indication of the primary source of unique 241Pu yield information in current and fu- ture experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The extreme 239Pu-241Pu yield offset in this example amplifies the impact of a modest non-constant relationship between F239 and F241 in LEU-based datasets, which is also illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To test the validity of this hypothesis with existing datasets, we perform a fit to only the RENO and Daya Bay LEU datasets while applying loose 75% external constraints on all four isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' While large un- certainty increases are seen in σ235 and σ238, σ239 and σ241 fractional uncertainties are altered by <30%, and fractional bounds on σ241 (43%) remain tighter than the 75% external constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, in the existing global dataset, it does ap- pear that that the Daya Bay and RENO LEU data points are responsible for the modest breaking of degeneracy between 239Pu and 241Pu yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Adding this to previously-established trends, it is straight- forward to recount the independent features of the global IBD yield dataset that enable determination of all four isotopes’ IBD yields: HEU-based experiments’ σf measurements directly constrain σ235 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The measured relative linear σf slope versus fuel burn-up at LEU-based experiments directly constrains σ239 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The time-integrated offset in σf between HEU and LEU cores constrains σ238 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The curvature of σf slope versus fuel burn-up at LEU experiments constrains σ241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As we move on to consider possible future IBD yield mea- surement scenarios, these high-level principles serve to guide attention toward those with particular promise for improving global knowledge of isotopic yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In particular, we will look to explore new multi-dataset measurements that can pro- vide an enhanced view of σf curvature with host reactor fuel evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We end this Section by noting that within the current global dataset, Daya Bay contains currently-unexploited potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [75] indicates O(5%) F241/F239 variations between re- actor cycles that are averaged out in its current fuel content binning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To estimate the achievable gains in the fis- sion yields, we generate an Asimov IBD yield dataset with fis- sion fractions taken from a combination of rates, RENO and Daya Bay-like experiment is divided into two halves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' one with the default fission fractions while the other having F241/F239 relatively reduced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The systematic and statistical un- certainties are assumed to match the existing global dataset and the yields are generated using best-fit results from the global dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Such a joint fit provides a modest improvement in the precision of fission yield of (σ235, σ238, σ239, σ241) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3%, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8%, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7%, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2%) compared to the precision of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3%, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2%, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%) for the existing global dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' If we further double the statistics of the Daya Bay Asimov data–as expected from the full Daya Bay dataset—in conjunc- tion with the modified binning in fission fractions, we find 7 0 1 2 3 4 5 6 7 8 /fission] 2 cm 43 [10 239 σ 0 2 4 6 8 10 12 14 16 18 20 22 /fission] 2 cm 43 [10 241 σ Correlation: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='990 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 235 σ 0 2 4 6 8 10 12 14 16 18 20 22 /fission] 2 cm 43 [10 241 σ Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='013 0 2 4 6 8 10 12 14 /fission] 2 cm 43 [10 238 σ 0 2 4 6 8 10 12 14 16 18 20 22 /fission] 2 cm 43 [10 241 σ Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='757 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Isotopic IBD yield fits for the existing global dataset with loose (75%) external constraints on the 241Pu IBD yield, σ241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Contours are pictured for σ241 relative to the other isotopic yields, with the fit marginalized over the non-pictured isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Correlation coefficients between fitted σ241 and the other yields are given in the plot legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Also shown in dashed lines are the theoretical IBD yields predicted by the Huber-Mueller model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Stars indicate IBD yields chosen for illustration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='95 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 /fission] 2 cm 43 IBD yield [10 Nominal IBD yields Modified IBD yields 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='75 235 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='24 239 /F 241 F Daya Bay RENO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Top: IBD yield sets for two hypothetical LEU measure- ments: one assuming measurements align with isotopic IBD yields matching the best-fit for the existing global dataset, and another as- suming alignment with σ239 and σ241 values matching those indi- cated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The latter scenario’s values lie outside of the 1 σ region preferred by the global IBD yield dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' for this scenario, σ238 is reduced to enable better vertical alignment of the two datasets and easier comparison of slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Bottom: Ratio (F241/F239) of the fission yields of 241Pu and239Pu for the hypothetical LEU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Realized F235 ranges for RENO and Daya Bay datasets are also pic- tured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' a further improvement in precision to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3%, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7%, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4%, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, we conclude that it may be worthwhile for Daya Bay to consider a more diversified fuel content binning scheme in a future analysis of its final full-statistics IBD yield dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This observation may also be applicable to other high- statistics datasets spanning many LEU reactor cycles, such as those recorded by RENO and DANSS [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' FUTURE IMPROVEMENTS FROM NEW MEASUREMENTS AT MULTIPLE CORE TYPES We now turn to consideration of future improvements in global knowledge of isotopic IBD yields by performing new measurements at a range of different reactor core types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We will begin by considering the most imminently-achievable next steps: short baseline measurements of a single LEU core over a full fuel cycle, and a subsequent systematically- correlated measurement at an HEU using the same νe de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We will then proceed to study possible improve- ments gained by making measurements at mixed-oxide and plutonium-burning fast reactor core types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Benefits of New HEU and LEU Measurements Some benefits of new measurements of IBD yields at short distances from a full LEU reactor core cycle have already been discussed in the literature [61], and have served as part of the physics motivation for the NEOS-II experiment [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In par- ticular, this configuration enables access to a wider range of F239 and F235 values beyond those achieved at θ13 exper- iments sampling multiple cores, which should result in im- proved σ239 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' When coupled with a systematically- correlated HEU-based measurement, which could be achieved 8 via two site deployments of the same detector system, di- rect constraints on σ238 may exceed the claimed precision of the summation prediction of Mueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Multi- ple current or near-future efforts, such as PROSPECT-II [79] or MAD [78], are well-suited to realize part or all of this com- bined LEU-HEU measurement program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Such a setup would also broaden access to LEU fuel content regimes with less linear relationships between F239 and F241, allowing for improved constraint of σ241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This improvement was demonstrated above for the hypothetical LEU measure- ments in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Realized effective F239 ranges for Daya Bay and RENO are also highlighted with shaded bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' we note that offsets in median F235 (and, while not pictured, also F241/F239) between hypothetical LEU and Daya Bay/RENO cases is due to the specifics of the single cycle core loading simulated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' A new short-baseline LEU measure- ment set can capture periods earlier and later in the fuel cycle of a conventional LEU core with respect to RENO and Daya Bay, when relative contributions of 239Pu and 241Pu fissions deviate most strongly from their cycle-integrated mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For the hypothetical short-baseline LEU measurement, F239/F241 varies roughly 6%, from 17% to 23%, over a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Daya Bay’s and RENO’s F241/F239 ratios, meanwhile vary by only 3% or less, with maximums and minimums of 20% and 17%, respectively [25, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The extent to which these HEU and LEU measurements can improve constraints on σ241 has so far not been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' To do so, we apply the four-parameter yield fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 5 to the hypothetical HEU and LEU datasets described in Section II B, Table I, and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Table II gives the result- ing precision in measurements of the four isotopic IBD yields probed by this new HEU+LEU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The most striking dif- ference with respect to the current global dataset is the sub- stantial improvement in knowledge of 239Pu and 241Pu yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Uncertainties in σ239 and σ241 are improved from 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% in the existing dataset to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5%, respectively, greater than four-fold improvement in both values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As illus- trated in Figure 5, this improvement can be partially attributed to the reduction in degeneracy between these two isotopes’ fis- sion fraction variations over a full LEU fuel cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' If all mea- surements are instead performed with a 1 ton detector, more closely approximating the expected size of the MAD detec- tor, uncertainties are similar in size, with σ235,238,239,241 shift- ing from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5%) for the PROSPECT-II sized detector case to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='62%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%) for the MAD detector case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, the HEU+LEU deployment sce- nario may yield major benefits for both physics-oriented or smaller applications-oriented future detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [61], σ238 constraints are also significantly improved, primarily due to the correlated nature of the detec- tor systematics assumed between the HEU and LEU measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' If this correlation is removed, or if the chosen opti- mistic 1% HEU thermal power uncertainties are increased to the currently-achievable 2% level, precision in knowledge of the 238U yield is substantially reduced – to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1% and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2% for these two cases, respectively – while precision in knowl- edge of the 241Pu yield is virtually unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, follow- ing the next generation of short-baseline HEU and LEU mea- surements, the precision of knowledge of the 241Pu yield may rival that of its sub-dominant 238U counterpart, and will be less dependent on a detailed understanding of host reactors’ thermal powers and on movement-induced changes in detec- tor response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' At this point, direct νe -based measurements of 241Pu fission attributes may begin to have useful application in testing the general accuracy of nuclear data knowledge for this isotope – similar to the value provided by νe -based con- straints of 238U from the current global dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Benefits from MOX Reactor Measurements Reactors burning mixed-oxide (MOX) fuels are another promising venue for performing IBD yield measurements with unique Fi combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In particular, the RG-MOX measurement case may be an imminently realizable one, given the presence and operation of RG-MOX commercial cores in Europe and Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The 50% reactor-grade mixed-oxide (RG- MOX) core described in Section II B features F239 far higher than an LEU core and broad variations in F241 from nearly 15% at reactor start-up to roughly 25% after one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Ra- tios F239/F241 vary much more widely from cycle beginning (27%) to end (45%) compared to the LEU reactor case above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Amidst these substantial fission fraction variations, 238U frac- tions remain relatively consistent between LEU and RG-MOX cases, offering further opportunity for reduction in degeneracy between 238U and the other isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Addition of a hypothetical ten-datapoint IBD yield dataset from this RG-MOX reactor core provides substantial enhance- ments in IBD yield precision when added to those of the short- baseline HEU and LEU datasets, which are also summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Expected precision of yields σ239 and σ241 are im- proved by another factor of ∼ 2 and ∼ 3 respectively when the hypothetical RG-MOX is added to the fit alongside the hypo- thetical HEU and LEU datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Meanwhile, σ238 yield preci- sion is also tightened to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7% expected relative uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Correlations between yield fit parameters for this case are also pictured in Figure 5, and appear further reduced between 239Pu and 241Pu with respect to the hypothetical HEU+LEU case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' As with the HEU+LEU case, if measurements are per- formed instead with a MAD-sized 1 ton detector target, only modest degradation in precision is seen: σ235,238,239,241 un- certainties shift from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4%) for a 4 ton target to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6%, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9%) for a 1 ton target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' On the other hand, if the correlation between the reactor measurements are removed, or if the chosen opti- mistic 1% HEU thermal power uncertainties are increased to the currently-achievable 2% level, precision in knowledge of the 238U and 241Pu yields are reduced—to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9%, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0% respectively— and are moderately worse than the theoretical yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Comparing this with the HEU+LEU case where the precision achievable on 238U yield is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1%, the improvement provided by the addition of RG-MOX reactor data doesn’t fully compensate for the loss in precision due to the lack of correlation or a reduction in thermal power uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' With measurements at three reactor types – HEU, LEU, and 9 Case Description Precision on σi (%) 235U 238U 239Pu 240Pu 241Pu Existing Global Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 1 HEU + LEU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 3 HEU + LEU + RG-MOX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 2 HEU + LEU + WG-MOX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 4 HEU + LEU + Fast 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 5 All 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 6 All, Uncorrelated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 Model Uncertainty [66] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Constraints on IBD yields of 235U, 238U, 239Pu, 240Pu, and 241Pu, from future hypothetical datasets from LEU and HEU reactors, given as a percentage of the best fit yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' For all cases unless noted, detector systematic uncertainties are assumed to be correlated between measurements, and a 75% external constraint is used for 241Pu and for 240Pu when applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The ‘All’ case considers inclusion of HEU, LEU, RG-MOX, VTR and PFBR yield measurements employing the same detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Model prediction uncertainties from [66] are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 /fission] 2 cm 43 [10 235 σ 7 8 9 10 11 12 13 14 /fission] 2 cm 43 [10 238 σ Correlation: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='383 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 /fission] 2 cm 43 [10 235 σ 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 239 σ Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='772 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 /fission] 2 cm 43 [10 235 σ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 241 σ Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='727 7 8 9 10 11 12 13 14 /fission] 2 cm 43 [10 238 σ 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 239 σ Correlation: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='511 7 8 9 10 11 12 13 14 /fission] 2 cm 43 [10 238 σ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 241 σ Correlation: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='681 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 239 σ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 /fission] 2 cm 43 [10 241 σ Correlation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='490 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Isotopic IBD yield contours for a combined fit of hypothetical HEU, LEU, and RG-MOX datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In each panel, fits are marginalized over the undepicted isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Correlation coefficients between each pair of isotopes are provided in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' MOX – with a common detector, direct IBD-based constraints on νe production by the four primary fission isotopes may be expected to rival or exceed the precision of conversion- based predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Most of these direct isotopic yield uncer- tainties are also smaller and more well-defined in origin than the O(5%) uncertainty attributed to summation predictions for these isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, with a global HEU+LEU+MOX dataset, one could generate IBD-based reactor νe flux predictions for many existing or future reactor types free from biases known to be present in conversion-predicted models without sacrific- ing relative model precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Expected isotopic IBD yield measurement precision de- livered by instead combining a ten datapoint weapons-grade mixed-oxide (WG-MOX) measurement with the hypothetical HEU and LEU datasets has also been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' IBD yield uncertainties for a HEU+LEU+WG-MOX measurement set are slightly worse than a HEU+LEU+RG-MOX set for σ238, σ239, and σ241 as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Similarity in results be- tween MOX fuel types should not be too surprising, since both WG-MOX and RG-MOX cycles roughly span a ∼ 16−17% 10 range in F239/F241 fission fraction ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' It is worth noting that wide variations in F239/F241 should also expected to be provided by conventional LEU cores burn- ing entirely fresh fuel, such as would occur upon first oper- ation of a new commercial power plant [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In this case, F239/F241 fission fraction ratios should be expected to vary by well over 10% over course of a fuel cycle [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Thus, in lieu of MOX-based options, IBD yield measurement regimes including newly started commercial cores likely serve as an- other promising avenue for producing precise constraints on all main fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Benefits from Fast Reactor Measurements Since fast fission cross-sections of many minor actinides – particularly 240Pu – are substantially higher than ther- mal fission cross-sections, fission fractions in the VTR and PFBR fast reactors are substantially different than those of the high-MOX-fraction conventional core configurations de- scribed in [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' In particular, 240Pu fissions now compose a non-negligible fraction of the total, and, as a result, 241Pu fis- sion fractions are substantially lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The addition of the two fast reactor dataset to the hypothetical HEU and LEU datasets is also summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The most striking product of introducing these datasets to the fit is the potential for set- ting the first-ever meaningful constraints on νe production by 240Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We find roughly comparable 240Pu yield measure- ments when either VTR or PFBR are fitted separately with the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Such a measurement could prompt new and deeper study of fission yields and decay data for this minor actinide, which plays a major role in the operation of next- generation fast reactor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The level of achievable preci- sion in the σ240 measurement is primarily driven by the preci- sion in understanding the thermal output of these fast reactor cores – an instrumentation challenge under active investiga- tion in the nuclear engineering community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Inclusion of fast reactor datasets generates only minor im- provements in the knowledge of σi for the other primary fission isotopes beyond that achievable with the HEU+LEU measurement scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' While this results primarily from the general lack of knowledge of the value of σ240, it also high- lights the value delivered by multiple highly systematically correlated measurements at differing fuel composition, like that provided by the MOX reactor cases, in contrast to the sin- gle measurement provided by the relatively static composition of these fast reactor cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Were F240 to evolve in a meaning- ful way for either core, it is likely that the isotopic IBD yield knowledge delivered by this core would be substantially im- proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' DISCUSSION AND SUMMARY After observing that the current global IBD yield dataset exhibits some capability to constrain antineutrino production by 235U, 238U, 239Pu, and 241Pu, we have investigated how suites of future systematically-correlated measurements at di- verse reactor core types can improve knowledge for these and other fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We have observed that with the simplest combination of correlated HEU and LEU measure- ments using a PROSPECT-sized or MAD-sized IBD detec- tor, an IBD yield measurement precision of 12% or better can be achieved for all four fission isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' With a combina- tion of HEU, LEU, and RG-MOX datasets, all isotopic yields can be directly measured with a precision rivaling or exceed- ing the precision claimed by conversion-predicted models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' If measurements of fast reactors are also included in the global dataset, first constraints of order 25% precision can be placed on antineutrino production by 240Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Beyond future measure- ments, we also noted other avenues for improving knowledge of isotopic IBD yields with current data: in particular, mea- surements performed over multiple LEU fuel cycles, such as Daya Bay and DANSS, can benefit from exploiting known variations in 241Pu between cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' With a combined global dataset in hand from multiple reactor types, one can generate IBD-based reactor νe flux predictions for many existing or future reactor types free from biases known to be present in conversion-predicted models without sacrificing in relative model precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' If one considers the full suite of correlated HEU, LEU, RG- MOX and fast reactor measurements (the “All” scenario in Table II), the resultant data-based model would include (σ235, σ238, σ239, σ240, σ241,) uncertainties of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The correlation between these achievable directly-constrained uncertainties has also been calculated, and can be seen in Figure 6, alongside those of the Huber- Mueller model [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Besides representing the similar mag- nitudes in uncertainty, Figure 6 shows direct measurements’ reduced correlations between 235U, 239Pu, and 241Puwith re- spect to conversion predictions, which are primarily caused by the common experimental apparatus used at ILL for input fission beta measurements [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' This kind of direct and precise understanding of all of the major fission isotopes’ contributions to reactor antineutrino emissions would represent movement into an era of ‘preci- sion flux physics’ offering many potential pure and applied physics benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' On the applications side, it would enable unbiased, high-fidelity monitoring, and performing of robust case studies for, a broad array of current and future reactor types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Well-measured isotopic antineutrino fluxes could be compared to summation-predicted ones to provide enhanced benchmarking and improvement of nuclear data associated with the main fission isotopes and their daughters, as well as the first meaningful integral datasets for validating the nu- clear data of 240Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' These models and correlated datasets would allow for precise independent tests of each of the four IBD yield predictions provided by the Huber-Mueller model, enabling thorough investigation of the hypothesis that mis- modelling of one or more isotopes’ yields is responsible for the reactor antineutrino anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Precise and reliable IBD- based flux constraints would also improve the reach of be- yond standard model searches with signal-dominated coherent neutrino-nucleus scattering detectors [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Finally, by probing for persistent residual IBD yield deficits common to all iso- 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='3 235 U 238 U 239 Pu 240 Pu 241 Pu 235 U 238 U 239 Pu 240 Pu 241 Pu 15 − 10 − 5 − 0 5 10 15 20 25 Uncertainty [%] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='6 235 U 238 U 239 Pu 240 Pu 241 Pu 235 U 238 U 239 Pu 240 Pu 241 Pu 15 − 10 − 5 − 0 5 10 15 20 25 Uncertainty [%] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Left: Uncertainties in isotopic IBD yield measurements based on a hypothetical global dataset including HEU, LEU, RG-MOX, and fast reactor IBD yield measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Diagonal elements correspond to the uncertainty in isotopic yields given for the “All” case in Table II, while off-diagonal elements describe the correlations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' The values are extracted by taking the square root of the corresponding elements of the correlation matrix and are assigned a negative value where the correlations are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Full covariance matrices are provided in the supplementary materials accompanying this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Right: Uncertainties in IBD yields predicted by the Huber-Mueller model [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Since there are no theoretical models predicting σ240, we assign 100% uncertainty on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' topes with respect to conversion or summation models, the community can search for enduring hints of sterile neutrino oscillations, even in the presence of other confounding effects, such as neutrino decay or wave packet de-coherence [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We encourage the use of the forecasted flux uncertainty matrix provided above and in the supplementary materials as input for future physics sensitivity and use case studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' these ex- ercises would help to directly demonstrate the value of this achievable advance reactor neutrino flux knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was supported by DOE Office of Science, un- der award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' DE-SC0008347, as well as by the IIT College of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' We thank Anna Erickson, Jon Link, and Patrick Huber for useful comments and discussion, and Nathaniel Bowden and Carlo Giunti for comments on early manuscript drafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Bemporad, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content=' Gratta, and P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} +page_content='10359 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFPT4oBgHgl3EQflzWN/content/2301.13123v1.pdf'} diff --git a/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/2301.11458v1.pdf.txt b/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/2301.11458v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a713b03874e7e3326929129040707bf4236bf8a9 --- /dev/null +++ b/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/2301.11458v1.pdf.txt @@ -0,0 +1,1913 @@ +Superconductivity of anomalous pseudospin +Han Gyeol Suh1, Yue Yu1,2, Tatsuya Shishidou1, Michael Weinert1, P. M. R. Brydon3, and Daniel F. Agterberg1 +1Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53201, USA +2Department of Physics, Stanford University, 476 Lomita Mall, Stanford, CA 94305, USA and +3Department of Physics and MacDiarmid Institute for Advanced Materials and Nanotechnology, +University of Otago, P.O. Box 56, Dunedin 9054, New Zealand +In materials with both time-reversal (T) and inversion symmetry (I), superconductivity is formed +by pairing fermion pseudospin partners at momenta k and −k. Typically, pseudospin shares the +same symmetry properties as usual spin-1/2. Here we consider non-symmorphic materials with mo- +mentum space spin-textures that exhibit an anomalous pseudospin with different symmetry prop- +erties than usual spin-1/2. We provide a comprehensive list of space groups for which anomalous +pseudospin occurs on planes in momentum space and carry out a complete categorization and anal- +ysis of superconductivity for Fermi surfaces centered on all possible T, I invariant momenta (TRIM) +in these planes. We show that superconductivity from this anomalous pseudospin leads to a vari- +ety of unusual consequences for superconductivity including: extremely large Pauli limiting fields +and residual Knight shifts for pseudospin singlet superconductors; field induced pair density wave +states; field induced pseudospin singlet to pseudospin triplet transitions; fully gapped ‘nodal’ super- +conductors; and additional insight into the breakdown of Blount’s theorem for pseudospin triplet +superconductors. We apply our results to UPt3, BiS2-based superconductors, Fe-based supercon- +ductors, and paramagnetic UCoGe. +I. +INTRODUCTION +Momentum space spin-textures of electronic bands are known to underlie spintronic and superconducting properties of +quantum materials [1–3]. In the spintronics context, Rashba-like spin textures allow control of electronic spin through +applied electric fields [1, 3]. +In superconductors, these same spin textures lead to unusual and counter-intuitive +magnetic response, such as the robustness of spin-singlet superconductivity to applied magnetic fields, pair density +wave states, and singlet-triplet mixing [2]. While such spin-textures are common when inversion symmetry is broken, +it has been realized that these can occur when inversion symmetry is present. This has lead to the notion of hidden +spin-textures [4] and locally non-centrosymmetric superconductivity [5], where inversion related sectors each allow a +Rashba-like spin-texture due to the local inversion symmetry breaking. These spin-textures are of opposite sign on +the two sectors, so that global inversion symmetry is restored. These hidden spin-textures allows the novel physics +associated with spin-orbit coupling (SOC) to emerge even when inversion symmetry is not broken. It further allows +for new physics to emerge. One notable example is a field induced transition from an even-parity (pseudospin singlet) +to odd-parity (pseudospin triplet) observed in CeRh2As2 [6–9]. +Key to observing novel physics associated with these spin-textures in inversion symmetric materials, is that the +inversion related sectors are weakly coupled [5, 9–11]. Theoretical proposals for how to achieve this fall under two +approaches: the first is to tailor weak coupling between the inversion related sectors, for example by separating two +inversion symmetry related layers so that the interlayer coupling is weak [6]; the second is to exploit symmetries that +ensure that this inter-sector coupling vanishes. The symmetry based approach has been applied to points and lines +in momentum space. Examples include two-dimensional (2D) transition metal dichalcogenides near the K-point [12] +and non-symmorphic symmetries near the X −M line in BaNiS2 with space group 129 (P4/nmm) [10]. In these cases, +the only energy splitting between the inversion-related sectors is due to SOC - a situation conceptually similar to +materials with broken inversion symmetry, where the usual two-fold pseudopsin degeneracy is broken solely by SOC. +Here we generalize this symmetry based approach by identifying electronic band degeneracies that are split solely +by SOC in materials with both inversion, I, and time-reversal, T, symmetries. This requires bands that are at least +four-fold degenerate when SOC is ignored. Such band degeneracies are not generic and require symmetries beyond the +usual two-fold pseudospin (or Kramers) degeneracy that arises from TI symmetry. Here we focus on 2D momentum +planes, nodal planes, where this occurs. This is the largest region in momentum space for which the required four- +fold electronic degeneracies can appear when SOC is ignored. As discussed in a variety of contexts [13–16], such +nodal planes arise in non-symmorphic crystal structures. Here we provide a complete list of space groups for which +this occurs and provide symmetry based kp theories for all time-reversal-invariant momenta (TRIM) on these nodal +planes. As discussed later, many relevant superconductors exhibit Fermi surfaces near these TRIM. We find that the +SOC-split electronic states on these nodal planes generically exhibit a pseudospin that has a different symmetry than +that of usual spin-1/2 fermions (this generalizes a result we found for space group P4/nmm [9]). Here we name this +anomalous pseudospin and examine the consequences of this anomalous pseudospin on superconductivity. We find +that this anomalous pseudospin plays a central role on the superconducting magnetic response and on the properties +arXiv:2301.11458v1 [cond-mat.supr-con] 26 Jan 2023 + +2 +of spin-triplet superconductivity. Our results complement and provide further insight on earlier nodal and topological +classifications of superconductivity in non-symmorphic materials [17–21]. +In this paper we begin by defining anomalous pseudospin on nodal momenta planes, we then characterize all possible +symmetry based kp theories near TRIM points on these nodal planes. Using these kp theories, we analyse the magnetic +response and nodal excitations of superconducting states formed from anomalous pseudospin. We apply this analysis +to a series of materials that exhibit Fermi surfaces that lie on or near these nodal planes. More specifically we reveal +how anomalous pseudospin: explains critical fields that far exceed the Pauli field in BiS2-based materials [22] and the +observed magnetic response 3D Fe-based superconductors [23]; identifies which space groups and TRIM are ideal to +find a field induced even parity to odd parity transition akin to that observed in CeRh2As2 [7]; provides insight into +the gap symmetry of UPt3 [24]; and shines new light on re-entrant superconductivity in UCoGe [25]. +II. +ANOMALOUS PSEUDOSPIN: SYMMETRY ORIGIN +Our aim is to exploit symmetry to find nodal plane band degeneracies that are lifted solely by SOC. As discussed +below, once these band degeneracies are lifted, a two-fold pseudospin degeneracy will remain. We find that generically, +the pseudospin that results from this procedure does not share the same symmetry properties as usual spin 1/2 and +hence we name this anomalous pseudospin. +Pseudospin describes the two-fold Kramers degeneracy that arises at each momentum point k when the product of +time-reversal T and inversion I symmetries, TI, is present. The product TI is anti-unitary and for fermions satisfies +(TI)2 = −1, ensuring at least a two-fold degeneracy. It is often the case that this pseudospin behaves as spin-1/2 +under rotations [26]. However, when symmetries beyond TI are present, it is possible that this is not the case. One +example of this is the angular momentum jz = ±3/2 electronic states that arise when cubic symmetry or a three-fold +rotation axis is present [2, 27, 28]. In the latter case, this gives rise to so-called type-II Ising superconductivity in 2D +materials [28, 29] where large in-plane critical fields appear when the Fermi surface is sufficiently close to momentum +points with this three-fold rotation symmetry. In our case, the anomalous pseudospin appears on momentum planes +in the Brillouin zone, allowing a larger phase space for the physical properties of anomalous pseudospin to manifest. +To ensure the requisite band degeneracy on a nodal plane, consider the symmetry elements that keep a momentum +point on the plane invariant (here taken to be normal to the ˆn axis). These are {E, ˜ +Mˆn, TI, T ˜C2,ˆn}, where +˜ +Mˆn +is a translation mirror symmetry and ˜C2,ˆn is a translation two-fold rotation symmetry. Their point group rotation +and translation component can be denoted using Seitz notation, for example ˜ +Mˆn = {Mˆn|t1, t2, t3} where Mˆn is a +point group mirror symmetry along ˆn and (t1, t2, t3) is a fractional translation vector. Since we are searching for a +degeneracy that appears without SOC, we consider orbital or sublattice degrees of freedom for which (TI)2 = 1. The +only remaining symmetry that can enforce a two-fold degeneracy is T ˜C2,ˆn, since this is anti-unitary, it must satisfy +(T ˜C2,ˆn)2 = −1 to do so. Since T commutes with rotations, this implies ˜C2 +2,ˆn = −1. When operating on orbital or +sublattice degrees of freedom, ˜C2 +2,ˆn is typically 1, suggesting it is not possible to have the required degeneracy. However, +in non-symmorphic groups, ˜C2,ˆn can be a screw axis, for which it is possible to satisfy ˜C2 +2,ˆn = −1. In particular, using +Seitz notation ˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (here t1 and t2 correspond to either a half in-plane translation vector or to no +translation) we have ( ˜C2,ˆn)2 = {E|0, 0, 1}. When operating on a state carrying momentum k, ( ˜C2,ˆn)2 is represented +by eik·ˆn. Hence if the nodal plane sits at momentum k · ˆn = π, then ˜C2 +2,ˆn = −1 and a two-fold orbital or sublattice +degeneracy is ensured. When spin-degeneracy is also included, these states are then four-fold degenerate when SOC +is ignored. +When SOC is included, it is possible to show that the TI pseudospin partners have the same Mˆn mirror eigenvalue +(this result is generalization of that given in Ref. [9] where t1 = 0 and t2 = 0 was used). That is, labeling the two +Kramers degenerate states as |+⟩ and TI|+⟩, both belong to the same eigenstate of ˜ +Mˆn. As a consequence, all Pauli +matrices ˜σi made from the two states |+⟩ TI|+⟩ must all be invariant under ˜ +Mˆn. It is this feature that differs from +usual spin-1/2. Of the three Pauli matrices σi, constructed from usual spin-1/2 states, two will be odd under ˜ +Mˆn and +one will be even under ˜ +Mˆn. It is this symmetry distinction between the anomalous pseudospin operators (˜σi) and +usual spin 1/2 operators (σi) that underlie the unusual superconducting properties discussed below. +The above argument can also be applied to nodal lines generated by the symmetry elements {E, ˜C2,ˆn, TI, T ˜ +Mˆn} +with (T ˜ +Mˆn)2 = −1 when applied to orbital or sublattice degrees of freedom. +In this case, repeating the same +arguments above show that SOC will also split the band degeneracy and lead to anomalous pseudospin. Here, due +to the larger available momentum phase space, we restrict our analysis and classification to nodal planes and leave +an analysis of nodal lines to a later work. For all space groups that host nodal planes, we develop symmetry-based +kp theories valid near all TRIM on theses nodal planes. We emphasis these TRIM since Cooper pairs are formed by +pairing states at momenta k and −k with the momentum origin given by a TRIM. We then consider Fermi surfaces + +3 +Z +E +B +D +Y +C +A +Γ +FIG. 1. +Example from space group 14 where the green shading reveals the planes and lines in momentum space on which +anomalous pseudospin exists. A Fermi surface located near the momentum plane kz = π (as depicted by the dark Fermi surface +near the Z point) will have its superconducting properties governed by pairing of anomalous pseudospin. However, Fermi +surfaces far from these planes (such as that depicted near the Γ point) will exhibit more usual superconducting properties. +near these TRIM and discuss the resultant superconducting properties. Figure 1 illustrates our approach. Here, in +green, we show the nodal planes and lines that exhibit anomalous pseudospin. Here we examine the properties of +superconductivity for a Fermi surface near the Z point, which is a TRIM on the nodal plane. The properties of +superconductivity for a Fermi surface near the Γ point, for which pseudospin is typically not anomalous, are described +in earlier review articles [30, 31]. We note that many superconducting materials, including the examples discussed in +this paper, exhibit Fermi surfaces near nodal planes. +III. +NODAL PLANE SPACE GROUPS AND SINGLE-PARTICLE kp HAMILTONIANS +Here we identify all space groups that allow anomalous pseudospin on nodal planes and construct the corresponding +symmetry-based kp-like Hamiltonians for all TRIM on these planes. +A. +Space groups with nodal planes +To identify these nodal planes, all space groups containing inversion symmetry I = {I|0, 0, 0} and the screw axis +˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (where t1 = 0, 1/2 and t2 = 0, 1/2) were identified. For these space groups, the nodal planes +lie on the Brillouin zone boundary. Table 1 lists the resultant space groups, point groups, nodal planes, and types +of kp theories allowed for these space groups. As discussed in the previous section, the degeneracies of these nodal +planes is generically lifted by SOC, yielding anomalous pseudospin. +B. +Symmetry based kp theories near TRIM +To understand the consequences of anomalous pseudospin on superconductivity requires a theory for the normal +state. Cooper pairs rely on the degeneracy between states of momenta k and −k and this degeneracy is ensured by +both T and I symmetries. For this reason, we develop symmetry-based kp theories expanded around TRIM. To derive +these kp-like Hamiltonians, we have used the real representations for the TRIM given in the Bilbao Crystallographic + +4 +Crystal Type +Number +Name +Nodal planes +kp theory classes +Monoclinic (C2h) +11 +P21/m +(u, 1/2, w) +Ctype1 +2h,1 +14 +P21/c +(u, 1/2, w) +Ctype1 +2h,1 , Ctype2 +2h,2 +Orthorhombic (D2h) +51 +Pmma +(1/2, v, w) +Dtype1 +2h,3 +52 +Pnna +(u, 1/2, w) +Dtype1 +2h,3 , Dtype2 +2h,4 , 8-fold +53 +Pmna +(u, v, 1/2) +Dtype1 +2h,3 , Dtype2 +2h,4 +54 +Pcca +(1/2, v, w) +Dtype1 +2h,3 , 8-fold +55 +Pbam +(1/2, v, w), (u, 1/2, w) +Dtype2 +2h,2 , Dtype1 +2h,3 +56 +Pccn +(1/2, v, w), (u, 1/2, w) +Dtype1 +2h,1 , Dtype2 +2h,2 , Dtype1 +2h,3 , 8-fold +57 +Pbcm +(u, v, 1/2), (u, 1/2, w) +Dtype1 +2h,3 , 8-fold +58 +Pnnm +(1/2, v, w), (u, 1/2, w) +Dtype1 +2h,1 , Dtype2 +2h,2 , Dtype1 +2h,3 , Dtype2 +2h,4 +59 +Pmmn +(1/2, v, w), (u, 1/2, w) +Dtype1 +2h,1 , Dtype1 +2h,3 +60 +Pbcn +(1/2, v, w), (u, v, 1/2) +Dtype1 +2h,3 , Dtype2 +2h,4 , 8-fold +61 +Pbca +(1/2, v, w), (u, v, 1/2), (u, 1/2, w) +Dtype1 +2h,3 , 8-fold +62 +Pnma +(1/2, v, w), (u, v, 1/2), (u, 1/2, w) +Dtype1 +2h,1 , Dtype1 +2h,3 , 8-fold +63 +Cmcm +(u, v, 1/2) +Ctype1 +2h,1 , Dtype1 +2h,3 +64 +Cmce +(u, v, 1/2) +Ctype2 +2h,2 , Dtype1 +2h,3 +Tetragonal (D4h) +127 +P4/mbm +(u, 1/2, w) +Dtype1 +2h,3 , Dtype2 +4h,2 , Dtype2 +4h,4 +128 +P4/mnc +(u, 1/2, w) +Dtype1 +2h,3 , Dtype2 +2h,4 , Dtype2 +4h,2 , Dtype2 +4h,4 , Dtype1 +4h,5 , 8-fold +129 +P4/nmm +(u, 1/2, w) +Dtype1 +2h,3 , Dtype1 +4h,1 , Dtype1 +4h,3 +130 +P4/ncc +(u, 1/2, w) +Dtype1 +2h,3 , Dtype1 +4h,1 , Dtype1 +4h,3 , 8-fold +135 +P42/mbc +(u, 1/2, w) +Dtype1 +2h,3 , Dtype2 +4h,2 , Dtype2 +4h,4 , 8-fold +136 +P42/mnm +(u, 1/2, w) +Dtype1 +2h,3 , Dtype2 +2h,4 , Dtype1 +4h,1 , Dtype2 +4h,2 , Dtype1 +4h,3 , Dtype2 +4h,4 +137 +P42/nmc +(u, 1/2, w) +Dtype1 +2h,3 , Dtype1 +4h,1 , Dtype1 +4h,3 , Dtype1 +4h,5 , 8-fold +138 +P42/ncm +(u, 1/2, w) +Dtype1 +2h,3 , Dtype1 +4h,1 , Dtype2 +4h,2 , Dtype1 +4h,3 , Dtype2 +4h,4 , 8-fold +Hexagonal (C6h) +176 +P63/m +(u, v, 1/2) +Ctype1 +2h,1 , Ctype1 +6h +, 8-fold +Hexagonal (D6h) +193 +P63/mcm +(u, v, 1/2) +Dtype1 +2h,3 , Dtype1 +6h +, 8-fold +194 +P63/mmc +(u, v, 1/2) +Dtype1 +2h,3 , Dtype1 +6h +, 8-fold +Cubic (Th) +205 +Pa3 +(u, 1/2, w) +Dtype1 +2h,3 , 8-fold +TABLE I. Space groups with nodal planes +server [32–34]. For these TRIM, we initially consider space group irreducible representations that do not include spin, +which, for simplicity, we name orbital representations. These representations are either 2-fold or 4-fold degenerate +(when spin is added, these becomes 4-fold and 8-fold degenerate respectively). The full kp-like Hamiltonians are only +listed for the 2-fold degenerate representations. We present a partial classification of the 4-fold degenerate orbital +representations near the end of this paper. +In constructing the kp theories for the 2-fold orbital degenerate TRIM points, we choose τi to be Pauli matrices that +encode the orbital degrees of freedom, and σi to be spin Pauli matrices. We take T = τ0(iσy)K where K is the complex +conjugation operator, hence the τ2 operator is odd under time-reversal. For a given doubly degenerate space group +representation on a TRIM, constructing its direct product leads to four irreducible point group representations. These +four representations each correspond to an orbital operator τi, and this partially dictates the momentum dependencies +of symmetry allowed terms in the kp Hamiltonian. We present our results for the kp Hamiltonians in Table 2. The +first row of each box gives the type of the kp theory class and the point group representations of the orbital operators +that are given by Pauli matrices τi. In this decomposition, the square brackets correspond to the antisymmetric τ2 +operator and remaining terms correspond to τ0, τ1, and τ3. The second row of a box gives the kp Hamiltonian, and +the last part of a box lists the space groups and TRIM points representations that belong to the kp Hamiltonian class. +We have tabulated the kp Hamiltonians for 122 TRIM points and we find that only 13 different kp theories appear. +These are of two types, which we call type 1 and type 2. Type 1 kp theories have degenerate even and odd parity +orbital basis functions. Type 2 kp theories has two degenerate orbital basis functions with the same parity symmetry. +The generic form of these kp theories are +H(k) = ε0,k + t1,kτ1 + tα,kτα + τβ(λk · σ) = ε0,k + Hδ(k) , +(1) + +5 +(I, τα, τβ) = +� +(τ1, τ2, τ3) +for type 1 , +(τ0, τ3, τ2) +for type 2 , +(2) +where Hδ(k) = H(k) − ε0,k and α and β are type indices will be used the remaining context. For parity mixed, type +1, kp theories, the degeneracy at TRIM points is not broken by SOC. This is because the non-symmorphic symmetries +combined with topological arguments imply these TRIM must have an odd number of Dirac lines passing through +them [35]. These Dirac lines lie in the nodal plane. Elsewhere in the nodal plane, SOC lifts the 4-fold degeneracy. +We will discuss some consequences of these Dirac lines later. The non trivial inversion symmetry for type 1, I = τ1, +implies the parity of the momentum functions that ε0,k = ε0,−k, t1,k = t1,−k, t2,k = −t2,−k, and λk = −λ−k. +This form of Hamiltonian has often been used to understand locally non-centrosymmetric superconductors [2] and +hidden spin polarization in inversion symmetric materials [11]. +In these contexts, the orbital degrees of freedom +reside on different sectors that are related by inversion symmetry and there is typically no symmetry requirement +that ensures the SOC dominates. The τ3 matrix is odd under inversion symmetry, allowing the odd-parity SOC +λk to appear. Many superconductors of interest have Fermi surfaces near type 1 TRIM points, examples include: +Fe-based superconductors, which often have electron pockets near the M point in space group 129 (classes Dtype1 +4h,1 +or +Dtype1 +4h,3 ) [23], in this context the high Tc superconductor monolayer FeSe is of interest, since it only has Fermi surfaces +near the M point [36]; CeRh2As2 which exhibits a field induced transition from an even parity to an odd-parity +superconducting state [7, 8] and has Fermi surfaces near the M point in space group 129 (classes Dtype1 +4h,1 +or Dtype1 +4h,3 ); +BiS2-based superconductors [22] which has superconductivity that survives to very high fields and which has electron +pockets near the X point in space group 129 (class Dtype1 +2h,3 ); the odd-parity heavy fermion superconductor UPt3 +[24] which has a pancake-like Fermi surface at kz = π/c in space group 193 (class Dtype1 +6h +); and the ferromagnetic +superconductor UCoGe [25] with space group 62 and a Fermi surface near the T point (class Dtype1 +2h,1 ). +For type 2 kp theories, the 4-fold degeneracy is sometimes already split into 2 at the TRIM point when SOC is +added, unlike what occurs for type 1 kp theories. This happens in classes Ctype2 +2h,2 +and Dtype2 +2h,1 . For the other type 2 +classes, this degeneracy at the TRIM point is not split. In these cases, an even number of Dirac lines pass through +the TRIM point. These Dirac lines lie in the nodal plane. Since I = τ0 for type 2, all terms in the Hamiltonian are +even parity, that is, unchanged under k → −k. One example where type 2 kp theories apply is in strain induced +superconductivity in RuO2[37, 38]. Without strain, RuO2 is thought to be a non-superconducting altermagnet [39]. +When strain is applied, bands near the X-M-R-A Brillouin zone face are most strongly affected [37]. RuO2 has space +group 136 with the R and M points belonging to classes Dtype2 +2h,4 , Dtype2 +4h,2 , or Dtype2 +4h,4 . Later we discuss the ferromagnetic +superconductor UCoGe with space group 62 [25]. In this example we highlight the role of 8-fold degenerate points +which exhibit some properties similar to that found for type 2 TRIM points. +Type 1 and type 2 kp Hamiltonians share some common features that play an important role in understanding +the properties of the superconducting states. The first is that the non-symmorphic symmetry dictates that these +Hamiltonians are best described as two-band systems with eigenenergies given by +E±(k) = ε0,k ± +� +t2 +1,k + t2 +α,k + |λk|2 = ε0,k ± εδ,k , +(3) +where α is the type index in Eq. 2. The second feature is that both simplify dramatically on the nodal plane, where +only the coefficient functions ε0,k and λk·ˆn are non-vanishing (that is t1,k = t2,k = t3,k = |λk׈n| = 0). This property +is a direct consequence of the anomalous pseudopspin. The symmetry arguments discussed in the previous section +enforce this condition. In particular, for momenta on the nodal plane, the mirror operator through the nodal plane, +UM, takes the from UM = −iτβ(σ · ˆn). The requirement that these Hamiltonians obey time-reversal and inversion +symmetries and commute with UM lead to this simple form of the kp theories in the nodal plane. The final important +property of these kp Hamiltonians is that the SOC terms are often the leading order terms in the kp expansions, that +is, they appear with the lowest powers of ki. This is the case for classes Ctype2 +2h,2 , Dtype1 +2h,1 , Dtype2 +2h,4 , Dtype1 +4h,2 , Dtype1 +4h,3 , and +Dtype1 +4h,5 . This feature ensures that there exists a limit in which the SOC is the dominant single-particle interaction on +the Fermi surface and hence the unusual magnetic superconducting response we later discuss must exist. +IV. +SUPERCONDUCTING STATES +In the previous section, complete symmetry-dictated kp theories were found for anomalous pseudospin. +These +theories are complete in the sense that they include all operators of the form τiσj allowed by symmetry. For super- +conductivity, the orbital degree of freedom enlarges the corresponding space of possible gap functions compared to +the usual even-parity (pseudospin-singlet) ˜∆(k) = ψk(iσy) and odd-parity (pseudospin-triplet) ˜∆(k) = dk · σ(iσy) + +6 +Class +Symmetry +Hamiltonian +Space Group Momenta +Ctype1 +2h,1 +Ag + Bg + [Au] + Bu +H = ϵ0 + (t1xkx + t1zkz)kyτ1 + t2kyτ2 ++ τ3[λxkyσx + (λyxkx + λyzkz)σy + λzkyσz] +11(C1, D1, E1, Z1), 14(Z1), +63(R1(yz)), 176(L1(yz)) +Ctype2 +2h,2 +Ag + 2Bg + [Ag] +H = ϵ0 + (t1xkx + t1zkz)kyτ1 + (t3xkx + t3zkz)kyτ3 ++τ2[(λxxkx + λxzkz)kyσx + λyσy + (λzxkx + λzzkz)kyσz] +14(D± +1 D± +2 ), 64(R± +1 R± +2 (yz)) +Dtype1 +2h,1 +Ag + B1g + [Au] + B1u +H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2 ++ τ3[λxkyσx + λykxσy + λzkxkykzσz] +56(S1,2), 58(R1,2) +59(S1,2, R1,2), 62(T1,2(xz)) +Dtype2 +2h,2 +Ag + 2B1g + [Ag] +H = ϵ0 + t1kxkyτ1 + t3kxkyτ3 ++ τ2[λxkykzσx + λykxkzσy + λzkxkyσz] +55(S± +1 S± +2 , S± +3 S± +4 , R± +1 R± +2 , R± +3 R± +4 ) +56(R± +1 R± +2 , R± +3 R± +4 ), 58(S± +1 S± +2 , S± +3 S± +4 ) +Dtype1 +2h,3 +Ag + B2g + [B1u] + B3u +H = ϵ0 + t1kxkzτ1 + t2kzτ2 ++ τ3[λxkxkykzσx + λykzσy + λzkyσz] +51(X1,2, S1,2, U1,2, R1,2), 52(R1,2(xy), Y1,2(xyz)) +53(Z1,2(zyx), T1,2(zyx)), 54(X1,2, S1,2) +55(U1,2(yz), X1,2(yz), Y1,2(xyz), T1,2(xyz)) +56(X1,2, Y1,2(xy)) +57(S1,2(xyz), Y1,2(xyz), Z1,2(zyx), U1,2(zyx)) +58(X1,2(yz), Y1,2(xyz)) +59(X1,2, U1,2, T1,2(xy), Y1,2(xy)), 60(X1,2, Z1,2(zyx)) +61(X1,2, Y1,2(xyz), Z1,2(zyx)) +62(X1,2, Z1,2(xz), Y1,2(xyz)) +63(T1,2(zyx), Z1,2(zyx)), 64(T1,2(zyx), Z1,2(zyx)) +127(X1,2(xyz), R1,2(xyz)), 128(X1,2(xyz)) +129(X1,2(xy), R1,2(xy)), 130(X1,2(xy)) +135(X1,2(xyz), R1,2(xyz)), 136(X1,2(xyz)) +137(R1,2(xy), X1,2(xy)), 138(X1,2(xy)) +193(L1,2), 194(L1,2(xy)) +205(X1,2(xyz)) +Dtype2 +2h,4 +Ag + B1g + B3g + [B2g] +H = ϵ0 + t1kxkyτ1 + t3kykzτ3 ++ τ2[λxkxkyσx + λyσy + λzkykzσz] +52(T ± +1 ), 53(U ± +1 (yz), R± +1 (yz)) +58(T ± +1 , U ± +1 (xy)), 60(S± +1 (xy)) +128(R± +1 ), 136(R± +1 ) +Dtype1 +4h,1 +A1g + B2g + [A1u] + B2u +H = ϵ0 + t1kxkyτ1 + t2kxkykz(k2 +x − k2 +y)τ2 ++ τ3[λx(kxσy + kyσx) + λ3kxkykzσz] +129(M1,2, A1,2), 130(M1,2) +136(A3,4), 137(M1,2), 138(M1,2) +Dtype2 +4h,2 +A1g + 2B2g + [A1g] +H = ϵ0 + t1kxkyτ1 + t3kxkyτ3 ++ τ2[λx(kykzσx + kxkzσy) + λzkxky(k2 +x − k2 +y)σz] +127(M ± +1 M ± +4 , M ± +2 M ± +3 , A± +1 A± +4 , A± +2 A± +3 ) +128(M ± +1 M ± +4 , M ± +2 M ± +3 ), 135(M ± +1 M ± +4 , M ± +2 M ± +3 ) +136(M ± +1 M ± +4 , M ± +2 M ± +3 ), 138(A± +1 A± +4 , A± +2 A± +3 ) +Dtype1 +4h,3 +A1g + B2g + [B1u] + A2u +H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2 ++ τ3[λx(kxσy − kyσx) + λzkxkykz(k2 +x − k2 +y)σz] +129(M3,4, A3,4), 130(M3,4) +136(A1,2), 137(M3,4), 138(M3,4) +Dtype2 +4h,4 +A1g + A2g + B2g + [B1g] +H = ϵ0 + t1kxky(k2 +x − k2 +y)τ1 + t3kxkyτ3 ++ τ2[λx(kykzσx + kxkzσy) + λzkxkyσz] +127(M ± +5 , A± +5 ), 128(M ± +5 ) +135(M ± +5 ), 136(M ± +5 ), 138(A± +5 ) +Dtype1 +4h,5 +A1g + A2g + [B1u] + B2u +H = ϵ0 + t1kxky(k2 +x − k2 +y)τ1 + t2kxkykzτ2 ++ τ3[λx(kxσy + kyσx) + λzkxkykzσz] +128(A1,2), 137(A1,2) +Ctype1 +6h +Ag + Bg + [Au] + Bu +H = ϵ0 + (t1xkx(k2 +x − 3k2 +y) + t1yky(3k2 +x − k2 +y))kzτ1 ++ t2kzτ2 + τ3[λxkz(2kxkyσx + (k2 +x − k2 +y)σy) ++ (λzxkx(k2 +x − 3k2 +y) + λzyky(3k2 +x − k2 +y))σz] +176(A1) +Dtype1 +6h +A1g + B2g + [A2u] + B1u +H = ϵ0 + t1kxkz(k2 +x − 3k2 +y)τ1 + t2kzτ2 ++τ3[λxkz(2kxkyσx + (k2 +x − k2 +y)σy) + λzky(3k2 +x − k2 +y)σz] +193(A1,2), 194(A1,2(xy)) +TABLE II. Classification of kp theories. Subscript numbering of momenta represents different real representations on the same +momentum point, and a permutation of the axes is denoted by the cyclic notation. For example, 128(X1,2(xyz)) represents +that there are two representations X1 and X2 on X = (0, 1/2, 0) space group 128, and their local theory is obtained by +Dtype1 +2h,3 +Hamiltonian under x → y → z → x relabelling. The representation convention is following Bilbao Crystallographic +servera[32–34] except for the L point in 193 and 194. +a https://www.cryst.ehu.es/ Representations and Applications → Point and Space Groups → - Representations → SG Physically +irreducible representations given in a real basis + +7 +states that appear in single-band theories [30, 31]. Nevertheless, it is possible to understand some general properties +of the allowed pairing states. +To deduce the symmetry properties of possible pairing channels in this larger space of electronic states, it is useful +to define gap function differently than usual [40, 41]. In particular, we take +H = +� +i,j,k +Hij(k)c† +k,ick,j + 1 +2 +� +i,j,k +[∆ij(k)c† +k,i˜c† +k,j + h.c.]. +(4) +where i, j are combined spin and orbital indices, h.c. means Hermitian conjugate, ck(c† +k) is the Fermionic spin-half +particle creation(annihilation) operator, and ˜ck(˜c† +k) is the time reversed partner of ck(c† +k). In the usual formulation ˜c† +k,j +is replaced c† +−k,j which leads a different gap function ˜∆ij and to difficulties in interpreting the symmetry transformation +properties of this gap function [40, 41]. For a single-band, these new gap functions become ∆(k) = ψkσ0 for even- +parity and ∆(k) = dk · σ for odd-parity. The key use of Eq. 4 is that the ∆ij(k) transform under rotations in the +same way as the Hij(k), allowing the symmetry properties of the gap functions to be deduced. The disadvantage of +this approach is that the antisymmetry of the gap functions that follows from the Pauli exclusion principle is not as +readily apparent compared to the usual formulation [40, 41]. +Enforcing the Pauli exclusion principle leads to eight types of gap functions that generalize the pseudospin-singlet +and pseudospin-triplet of single-band gap functions. Six of these are simple generalizations of the single-band gap +functions: τiψk and τi(dk · σ) for i = 0, 1, and 3 where ψ−k = ψk and d−k = −dk. Two are new gap functions: +τ2(ψk · σ) and τ2dk with ψ−k = ψk and d−k = −dk. It is possible to determine whether these gaps functions are +either even or odd-parity and this depends upon whether the kp Hamiltonian is type 1 or type 2. These gap functions +and their parity symmetry are listed in Table III. Without further consideration of additional symmetries, the gap +function will in general be a linear combination all the even (or odd) parity gap functions. +To gain an understanding of the relative importance of these pairing states it is useful to project these gaps onto the +band basis. Such a projection is meaningful if the energy separation between the two bands is much larger than the gap +magnitude. For many of the kp Hamiltonians, due to the presence of Dirac lines, there will exist regions in momentum +space for which this condition is not satisfied. However, these regions represent a small portion of the Fermi surface +when the SOC energies are much larger than the gap energies, so that an examination of the projected gap is still +qualitatively useful in this limit. Provided the superconducting state does not break time-reversal symmetry, the +projected gap magnitude on band a can be found through [42] +˜∆2 +± = Tr[|{Hδ, ∆}|2P±] +Tr[|Hδ|2] +. +(5) +where P±(k) = 1 +2(1 ± Hδ(k)/εδ,k) which is a projection operator onto ± band by the energy dispersion Eq. 3. This +projected gap magnitude is related to superconducting fitness [43, 44]: if it vanishes, the corresponding gap function +is called unfit and will have a Tc = 0 in the weak coupling limit. Table III gives the projected gap functions for +the pairing states discussed above. The projection generally reduces the size of the gap, with the exception of the +usual even-parity τ0ψk state (interestingly, the odd-parity τ0(dk ·σ) state has a gap that is generically reduced). This +reduction strongly suppresses the Tc of the pairings state, where it enters exponentially in the weak-coupling limit. +We later examine the different kp classes to identify fit gap functions since the Tc of these states will be the largest, +given a fixed attractive interaction strength. +On the nodal plane, the projected gap functions, shown in Table III, simplify considerably since only ε0 and λk · ˆn +are non-zero. For both type 1 and type 2 Hamiltonians, this leads to two gap functions that are fully fit, that is, +not reduced by the projection. For type 1 Hamiltonians, these fully fit states are τ0ψk and τ3ψk. The state τ0ψk is +even-parity and the state τ3ψk is odd-parity and, as discussed later, these two states play an important role in the +appearance of a field-induced transition from even to odd parity superconductivity as observed in CeRh2As2. For +gap functions described by vectors, for example dk, the projected gaps on the nodal plane are of the form |dk · ˆn|2 +or |dk × ˆn|2. This is qualitatively different than the usual odd-parity single-band gap, where the gap magnitude is +|dk|2. The latter requires that all three components of dk must vanish to have nodes. For the projected gaps on the +nodal planes, this requirement less stringent: only one or two components of dk need to vanish to have nodes. This +is closely related to the violation of Blount’s theorem on the nodal planes. +A. +Gap projection and the violation of Blount’s theorem +Blount’s theorem states that time-reversal symmetric odd-parity superconductors cannot have line nodes when SOC +is present [40]. Key to Blount’s theorem is the assumption that pseudsopsin shares the same symmetry properties + +8 +Type 1 +Type 2 +Gap function Inversion +Gap projection +Gap on nodal plane Inversion +Gap projection +Gap on nodal plane +τ0ψ ++ +|ψ|2 +|ψ|2 ++ +|ψ|2 +|ψ|2 +τ0(d · σ) +− +(t2 +1 + t2 +2)|d|2 + |d · λ|2 +t2 +1 + t2 +2 + |λ|2 +|d · ˆn|2 +− +(t2 +1 + t2 +2)|d|2 + |d · λ|2 +t2 +1 + t2 +2 + |λ|2 +|d · ˆn|2 +τ3ψ +− +|λ|2|ψ|2 +t2 +1 + t2 +2 + |λ|2 +|ψ|2 ++ +t2 +3|ψ|2 +t2 +1 + t2 +3 + |λ|2 +0 +τ3(d · σ) ++ +|d · λ|2 +t2 +1 + t2 +2 + |λ|2 +|d · ˆn|2 +− +t2 +3|d|2 + |d × λ|2 +t2 +1 + t2 +3 + |λ|2 +|d × ˆn|2 +τ1ψ ++ +t2 +1|ψ|2 +t2 +1 + t2 +2 + |λ|2 +0 ++ +t2 +1|ψ|2 +t2 +1 + t2 +3 + |λ|2 +0 +τ1(d · σ) +− +t2 +1|d|2 + |d × λ|2 +t2 +1 + t2 +2 + |λ|2 +|d × ˆn|2 +− +t2 +1|d|2 + |d × λ|2 +t2 +1 + t2 +2 + |λ|2 +|d × ˆn|2 +τ2d ++ +t2 +2|d|2 +t2 +1 + t2 +2 + |λ|2 +0 +− +|λ|2|d|2 +t2 +1 + t2 +3 + |λ|2 +|d|2 +τ2(ψ · σ) +− +t2 +2|ψ|2 + |ψ × λ|2 +t2 +1 + t2 +2 + |λ|2 +|ψ × ˆn|2 ++ +|ψ · λ|2 +t2 +1 + t2 +3 + |λ|2 +|ψ · ˆn|2 +TABLE III. Classification of allowed pairing states for the kp theories. For both type I and II TRIMs we give the symmetry +under inversion, the gap projection onto the Fermi surface, and the gap on the nodal plane. The momentum subscript indices +k of the coefficient functions are omitted here. +as usual spin [40]. While the violation of Blount’s theorem in non-symmorphic space groups has been demonstrated +earlier [18, 20, 21], here we present a simple proof that closely links anomalous pseudopsin to the violation of Blount’s +theorem. +The existence of anomalous pseudospin requires the presence of the translation mirror symmetry ˜ +Mˆn. Consequently, +the gap function can be classified as even or odd under this symmetry. Momenta on the nodal plane are invariant +under ˜ +Mˆn. Hence, for these momenta, U † +M∆(k)UM = ±∆(k) where the + (−) holds for a mirror-even (mirror-odd) +gap function. For our basis choice UM = −iτβ(σ · ˆn). Importantly, for both types the kp theories on the nodal plane +are given by H(k) = ε0,k +iUM(λk · ˆn). This defines the two bands E±(k) = ε0,k ±|λk · ˆn|. Written in the band basis, +we can divide the pairing potential into intraband and interband components. On the nodal plane the intraband gap +functions are explicitly given by +P±∆P± = 1 +4(−UM ± i sgn(λk · ˆn)){UM, ∆} , +(6) +while the interband components are +P±∆P∓ = 1 +4(−UM ± i sgn(λk · ˆn))[UM, ∆] +(7) +We observe that since a mirror-even gap function satisfies [UM, ∆] = 0, the interband gap components must vanish +on the nodal plane, i.e. the pairing only involves particles from the same band. The general form of the BdG energy +dispersion relation is then +±′ � +(ε0,k ± |λk · ˆn|)2 + |∆±±|2 , +(8) +where intraband gap magnitude |∆±±|2 = +1 +4Tr[|P±∆P±|2] and ±′ is the particle-hole symmetry index which is +independent of band index ±. Since there is no requirement that |∆±±|2 = 0, line nodes are therefore not expected +on the nodal plane, but rather we should generically find two-gap behavior with different size gaps on the two bands. +In contrast, for the mirror-odd gap functions we have {UM, ∆} = 0, so there is no intraband pairing on the nodal +plane. The eigenenergies for this interband pairing state are then +±′ � +±|λk · ˆn| + +� +ϵ2 +0,k + |∆±∓|2 +� +, +(9) +where intraband gap magnitude |∆±∓|2 = 1 +4Tr[|P±∆P∓|2]. The gap has line nodes provided |λk · ˆn|2 > |∆±∓|2. This +result depends only on the mirror-odd symmetry of the gap, and not on the parity symmetry. Since gaps which are +odd under both mirror and parity symmetry are allowed, this result shows that odd-parity gaps can have line nodes, +thus demonstrating a violation of Blount’s theorem. + +9 +The origin of these nodes due to purely interband pairing implies that the nodes are shifted off the Fermi surface +[45]. If the spin-orbit coupling is too weak, i.e. |λk · ˆn|2 < |∆±∓|2, the nodes can annihilate with each other and are +absent. This possibility has been discussed in the context of monolayer FeSe [46] and UPt3 [47]. The analysis above +is valid even when Dirac lines pass through the TRIM points, as is the case in most of the derived kp theories. On +the Dirac lines, the condition |λk · ˆn|2 < |∆±∓|2 must occur and the spectrum is therefore gapped. In the Appendix +A we present exact expressions for the energy eigenstates on the nodal plane for all possible combinations of mirror +and parity gap symmetries. +B. +Unconventional pairing states from electron-phonon interactions +To highlight how pairing of anomalous pseudospin can differ from the single-band superconductivity, it is instructive +to consider an attractive U Hubbard model. Such a model is often used to capture the physics of electron-phonon +driven s-wave superconductivity in single-band models. Here we show that this coupling also allows unconventional +pairings states. In particular, odd-parity states in type 1 kp Hamiltonians. Such a state has recently likley been +observed in CeRh2As2. +Here we consider a local Hubbard-U attraction on each site of the lattice and do not consider any longer range +Coulomb interactions. These sites are defined by their Wyckoff positions. Importantly, for the non-symmorphic groups +we have considered here, each Wyckoff position has a multiplicity greater than one. Here we limit our discussion to +Wyckoff positions with multiplicity two, which implies that there are two inequivalent atoms per unit cell. +An +attractive U on these sites stabilizes a local spin-singlet Cooper pair. Since there are two sites per unit cell this +implies that there are two stable superconducting degrees of freedom per unit cell. These two superconducting states +can be constructed by setting the phase of Cooper pair wavefunction on each site to be the same or opposite. Since +only local interactions are included, both these two states will have the same pairing interaction. The in-phase state +is a usual s-wave τ0ψk state. Identifying the other, out of phase, superconducting state requires an understanding of +the relationship between the basis states for the kp Hamiltonians and orbitals located at the Wyckoff positions. In +general, this will depend on the specific orbitals included in the theory. However, the condition that the resultant +pairing states must be spin-singlet and local in space (hence momentum independent) allows only two possibilities +for this additional pairing state: it is either a τ1ψk or a τ3ψk pairing state. Of these states, for two reasons, the τ3ψk +state for type 1 Hamiltonains is of particular interest. The first reason is that this state is odd-parity and therefore +offers a route towards topological superconductivity [48, 49]. The second reason is that of the four possible states +(τ1ψk or τ3ψk for type 1 or type 2 Hamiltonians), this is the only state that is fully fit on the nodal plane (as can be +seen in Table III, the other three states have zero gap projection on the nodal plane). This implies that for type 1 +Hamiltonians, the odd-parity τ3ψk and the s-wave τ0ψk states can have comparable Tc since they both have the same +pairing interaction. In practice, the τ3ψk state will have a lower Tc than the τ0ψk state since it will not be fully fit away +from the nodal plane. Table III reveals that this projection is given by the ratio |λk|2/(t2 +1,k +t2 +2,k +|λk|2). For classes +Dtype1 +2h,1 , Dtype1 +4h,1 , Dtype1 +4h,3 , and Dtype1 +4h,5 , this ratio is nearly one since the SOC terms are the largest in the kp Hamiltonian. +This suggests that these classes offer a promising route towards stabilizing odd-parity superconductivity. We stress +that because |λk|2/(t2 +1,k + t2 +2,k + |λk|2) is slightly less than one, the Tc of the odd-parity τ3ψk will be comparable but +less than that of the usual s-wave state. However, as we discuss later, the τ3ψk state can be stabilized over the usual +s-wave τ0ψk state in an applied field. The identification of classes Dtype1 +2h,1 , Dtype1 +4h,1 , Dtype1 +4h,3 , and Dtype1 +4h,5 +that maximize +the Tc of odd-parity pairing from electron-phonon interactions allows the earlier theory for a field induced even to +odd parity transition CeRh2As2 [9] (with space group 129) to be generalized to many other space groups. +While the above odd-parity state is only relevant for type 1 Hamiltonians, for type 2 Hamiltonians, the usual s-wave +interaction can develop a novel structure. In particular, for the classes Ctype2 +2h,2 +and Dtype2 +2h,4 , Table II shows that the +state τ2σy is maximally fit and has s-wave symmetry. Consequently, this state will admix with the usual s-wave τ0ψ +state. The theory describing this admixture formally resembles that of a Hund pairing mechanism proposed to explain +the appearance of nodes in the likely s-wave superconductor KFe2As2 [50]. The results of this analysis and a follow +up analysis [51] allows some of the properties of this state to be understood. An important conclusion of these works +is that an s-wave superconducting state can emerge even when pairing for the usual s-wave state is repulsive (that is +for the Hubbard U > 0). This holds if two conditions are met: the effective interaction for the τ2σy state is attractive +(to first approximation, this effective interaction does not depend upon U [50, 51]) and the two bands that emerge in +the kp theory both cross the chemical potential. This s-wave pairing state naturally lead to nodes. + +10 +V. +ROLE OF MAGNETIC FIELDS +The role of anomalous pseudopsin is perhaps most unusual in response to magnetic fields. In many superconductors, +there has been a push to drive up the magnetic field at which these are operational. Ising superconductors are one +class of materials for which this has been successful, the in-plane critical field far surpasses the Pauli field, opening +the door to applications [52]. Another relevant example is the field induced transition from an even parity to an +odd-parity state observed in CeRh2As2 [7, 8]. +Recently, a powerful method to examine the response of superconductors to time-reversal symmetry-breaking fields +has been developed by the projection onto the band-basis[42]. The form of the kp theories we have developed allows +for the direct application of this projection method. The response of superconductivity to time-reversal symmetry- +breaking is described by a time-reversal symmetry-breaking interaction Hh(k). A common form of TRSB Hamiltonian, +and the one we emphasize here, is the Zeeman field interaction term, which is represented by +Hh(k) = τ0(h · σ) , +(10) +where h is a magnetic field parameter in the system. We note that our qualitative results apply to a broader range of +TRSB Hamiltonians. In particular, this is true if the TRSB field shares the same symmetry properties as a Zeeman +field (for example if Hh(k) describes the coupling between orbital angular momentum and an applied field). +The theory introduces two parameters that quantify the response of superconductivity to time-reversal symmetry- +breaking. The first parameter is an effective g-factor given by +˜g2 +±,k,h = 2Tr[|{Hδ, Hh}|2P±] +Tr[|Hδ|2]Tr[|Hh|2] . +(11) +The second parameter is the field-fitness, given by +˜F±,k,h = +Tr[|{{Hδ, ˜∆}, {Hδ, Hh}}|2P±] +2Tr[|{Hδ, Hh}|2P±]Tr[|{Hδ, ˜∆}|2P±] +. +(12) +This field-fitness function ranges in value from zero to one. When the field-fitness is zero, the superconducting state +is not suppressed by the time-reversal symmetry breaking perturbation. With these two parameters, the response of +superconductivity to applied fields and the temperature dependence of magnetic susceptibility in the superconducting +state can be determined. With the choice of the time-reversal symmetry-breaking field as the Zeeman field, Eq. 10, +one finds +˜g2 +±,k,h = +t2 +1,k + t2 +α,k + (λk · ˆh)2 +t2 +1,k + t2 +α,k + λ2 +k +(13) +where α is a type index that is 2 for type 1 and 3 for type 2. This agrees with results in [53] derived for Hamiltonians +that resemble type 1 Hamiltonians. We note that the band index ± and the magnitude of field h in the field-fitness +and the g-factor do not change the outcome, thus they will be omitted in the subsequent sections and they will be +denoted by ˜F 2 +k,ˆh and ˜g2 +k,ˆh. +A. +Even parity superconductors +It can be shown that the field-fitness parameter in Eq. 12 is 1 for all even parity states. Consequently, the magnetic +response is governed solely by the generalized g-factor given in Eq. 13. For momenta on the nodal plane, where +t1,k = t2,k = t3,k = λk × ˆn = 0, the g-factor vanishes for magnetic fields orthogonal to ˆn. This is a direct consequence +of the anomalous pseudospin, since the symmetries of the Pauli matrices formed from anomalous pseudospin do +not allow any coupling to a Zeeman field perpendicular to ˆn. An immediate consequence is that superconductivity +survives to much stronger fields than expected for these field orientations. However, momenta that do not sit on +the nodal plane also contribute to the superconducting state and their contribution needs to be included as well. To +quantify this, we solve for the Pauli limiting field within weak coupling theory at T = 0. For an isotropic s-wave +superconductor, we find +ln +hP,ˆh +h0 += −⟨ln |˜gk,ˆh|⟩k +(14) + +11 +-1.0 +-0.5 + 0.0 + 0.5 + 1.0 +Γ +X +M +(a) +Energy (eV) +-1.0 +-0.5 + 0.0 + 0.5 + 1.0 +Γ +X +M +(b) +FIG. 2. +DFT bands of BiS2 near the X point (a) without and (b) with the SOC. The bands highlighted in the box are our +focus. +for field along direction ˆh, where h0 is the usual Pauli limiting field (found when the SOC is ignored), and ⟨·⟩k means +an average over the Fermi surface weighted by the density of states. Below, we apply this formula to BiS2-based +superconductors. We note that the spin susceptibility in the superconducting state can also be expressed using ˜gk,ˆh +as well [42], and this shows that a non-zero spin susceptibility is predicted at zero temperature whenever the critical +field surpasses h0. +1. +Enhanced in plane field Pauli for BiS2-based superconductors +Here we turn to recent experimental results on BiS2-based superconductors [22, 54]. This material has the tetragonal +space group 129 (P4/nmm) and it exhibits two electron pockets about the two equivalent X points [55]. When S is +replaced with Se, it has been observed that the in-plane upper critical field surpasses the usual Pauli limiting field by +a factor of 7 [54]. While it has been suggested that the local non-centrosymmetric structure is the source of this large +critical field [54], there has been no quantitative calculation for this. Here we apply Eq. 14 to the kp theory at the +X-point to see if it is possible to account for this large critical field. The X point in space group 129 belongs to class +Dtype1 +2h,3 .For BiS2, the dispersion is known to be strongly two-dimensional (2D) [22, 55] so we consider the kp theory in +the 2D limit. This kp theory is +HBiS2 = ¯h2 +2m +� +k2 +x + γ2k2 +y +� +− µ + t2kyτ2 + λxkyτ3σx + λykxτ3σy. +(15) +Assuming s-wave superconductivity and accounting for the two equivalent pockets yields +hP,ˆx = h0 +� +t2 +2 + λ2x + |γλy| +� +|t2| + |γλy|(t2 +2 + λ2x)1/4 +(16) +where h0 is the usual Pauli limiting field. For simplicity we consider γ = 1 in the following. Eq. 16 reveals that a large +enhancement of the limiting field is possible and requires two conditions. The first is that t2 << λx, λy and second is +that these is substantial anisotropy in λx and λy. To understand if these conditions are reasonable, we have carried +out density-functional theory (DFT) calculations on LaO1/2F1/2BiS2 with and without SOC. DFT calculations for +LaO1/2F1/2BiS2 were carried out by the full-potential linearized augmented plane wave method [56]. The Perdew- +Burke-Ernzerhof form of the exchange correlation functional [57], wave function and potential energy cutoffs of 14 and +200 Ry, respectively, muffin-tin sphere radii of 1.15, 1.2, 1.3, 1.0 ˚A for Bi, S, La, O atoms, respectively, the experimental +lattice parameters [58], and an 15 × 15 × 5 k-point mesh were employed for the self-consistent field calculation. The +virtual crystal approximation was used by setting the nuclear charge Z = 8.5 at O(F) sites. The resultant bands are +shown in Fig. 2. Without SOC, the band splitting along Γ to X yields an estimate for t2. When SOC is present, the +band splitting along the X to M yields λy and the band splitting along Γ to X yields +� +λ2x + t2 +2. The DFT calculated +splittings suggest that λx is the largest parameter by a factor of 3-4, while t2 and λy are comparable. This suggests +that the conditions to achieve a large critical field are realistic in BiS2-based superconductors. Note that the largest + +12 +observed Pauli fields are found when the S is substituted by Se [54]. Se has a larger SOC than S, suggesting that the +λi parameters will be increased from what we estimate here. This is currently under exploration. +It is worthwhile contrasting the above theory with that for Fe-based materials in which electron pockets exist near +the M point of space group 129. The M-point is described by class Dtype1 +4h,1 . In this case, an analysis similar to to +BiS2 gives an enhancement of only +√ +2 of the Pauli field for in-plane fields. For c-axis fields, this class implies a +significantly enhanced Pauli limiting field. These results are consistent with experimental fits to upper critical fields +in Fe-based superconductors that reveal that the upper critical field for in-plane fields are Pauli suppressed while those +for field along the c-axis are not [59]. The contrast bewteen Fe-based materials and BiS2-based materials highlights the +importance of the different classes. In particular, the lower orthorhombic symmetry of the X point allows protection +to in-plane fields not afforded to the M point, where the theory is strongly constrained by tetragonal symmetry. +2. +Pair density wave states +In BCS theory, a spin-singlet superconductor is suppressed by the Zeeman effect. +Under a sufficiently strong +magnetic field, the pairing susceptibility can be peaked at non-zero Cooper pair momenta, leading to a pair density +wave or FFLO state [60–62]. A schematic phase diagram for a centrosymmetric system is shown in the left panel +of Fig.3. The typically first order phase transition (double solid line) between the uniform and FFLO state ends at +a bicritical point (Tb, Hb), i.e. FFLO state only exists for T < Tb. A weak-coupling calculation reveals that for the +usual FFLO phase, Tb/Tc = 0.56 +It is known that for locally non-centrosymmetric superconductors, FFLO-like phases can appear at lower fields Hb +and higher temperatures Tb than the usual FFLO-like instability [5]. This is closely linked to the symmetry required +instability to a pair density wave state for non-centrosymmetric superconductors when a field is applied [2]. For a +non-centrosymmetric system under magnetic field, both inversion and time-reversal symmetry are broken. As a result, +the pairing susceptibility is generically peaked at non-zero momentum and Tb = Tc. For locally non-centrosymmtric +superconductors, inversion symmetry is locally broken on each sublattice. In an extreme case, if the two sublattices +are decoupled, then the system effectively becomes non-centrosymmetric, and under a small magnetic field, an FFLO +state can exists right below the zero-field superconducting Tc. However, these sublattices are generically coupled so +that Tb = Tc is not realized in practice. Here we show that for type 1 Hamiltonians, FFLO-like states can in principle +exist up to Tb = Tc. +FIG. 3. Schematic phase diagram for a spin-singlet superconductor under Zeeman effect. Single solid lines denote continuous +phase transitions while double solid lines denote first-order phase transitions. +To show this, we consider the 2D version of class Dtype1 +4h,1 +and use the pairing susceptibility to calculate Tb and Hb. +In 2D, class Dtype1 +4h,1 +has the following normal state Hamiltonian: +HD4h,1 = ¯h2 +2m(k2 +x + k2 +y) − µ + t1kxkyτ1 + λxτ3(kyσx + kxσy) + Hxσx +(17) +λx denotes the strength of the local inversion symmetry breaking (local Rashba SOC), while t1 is the inter-sublattice + +13 +coupling. The pairing susceptibility for an s-wave state with gap function τ0ψk is +χpairing(Q) = − 1 +β +� +ωn +� +(p,p+Q)∈FS +Tr [G0(Q + p, ωn)G0(p, ωn)] , +(18) +where G0 is the normal state Green’s function written in Nambu space. The FFLO state is favored, if the pairing +susceptibility is peaked at non-zero Q. We examine the position of the bicritical point (Tb, Hb), as a function of +λx/(t1kF ). We use the following two equations to locate the bicritical point: (1) The bicritical point lies on the BCS +transition for the uniform superconductivity. (2) The bicritical point is a continuous phase transition between uniform +and FFLO superconductivity, where ∇2 +Qχpairing(Q) = 0. The result is in Fig. 4. 1000 × 1000 points are sampled in +the 2D Brillouin zone. Other parameters are t1 = 0.2, t = µ = 1. An energy cutoff of Ec = 0.1 is applied to determine +the position of the Fermi surface. +0 +0.02 +0.04 +0.06 +0.08 +0.1 +x/t 1/kF +0 +0.2 +0.4 +0.6 +0.8 +1 +Tb/Tc +0 +0.02 +0.04 +0.06 +0.08 +0.1 +x/t 1/kF +0 +0.2 +0.4 +0.6 +0.8 +1 +Hb/Hb( x=0) +FIG. 4. The position of the bicritical point (Tb, Hb), as a function of λx/kF t1. +These results show that for zero λx/kF t1, a usual FFLO phase is found (that is Tb/Tc ≈ 0.56). As the SOC λx +increases or equivalently, as kF decreases, Tb increases and approaches the zero-field critical temperature. In the +meantime, Hb monotonically decreases. +We have shown that the FFLO phase can exist up to Tb = Tc for a 2D version of class Dtype1 +4h,1 . Key is that SOC is +the leading order term in the kp theory and this is also the case for other type 1 Hamiltonians. Hence the optimal +conditions for an enhanced FFLO phase to occur are when fields are applied in-plane (perpendicular to the c-axis) +for classes Dtype1 +2h,1 , Dtype1 +4h,1 , Dtype1 +4h,3 , and Dtype1 +4h,5 . +B. +Odd-parity superconductors +For odd parity superconductors, the field fitness parameter ˜Fk,ˆh can become less than 1 [42]. Of particular interest +is when ˜Fk,ˆh = 0 since this implies that Tc is unchanged by the time-reversal symmetry breaking field (this is +independent of the effective g-factor) [42]. For anomalous pseudospin this possibility leads to two consequences not +expected for spin-triplet states made from usual spin-1/2 fermions. The first is a field induced transition from an +even to an odd parity state. The second is that, in spite of the presence of strong SOC, the superconducting state is +immune to magnetic fields for all field orientations. We discuss these each in turn. +1. +Field induced even to odd parity transitions +In CeRh2As2, a field induced even to odd parity transition has been observed for the field oriented along the c-axis +in this tetragonal material [7, 8]. Earlier, we argued that this was due the anomalous pseudospin that arises on the +Brillouin zone faces in the non-symmorphic space group P4/nmm [9]. Here we show how this can be generalized +to other space groups that admit type 1 kp theories and determine which classes are optimal for observing such a +transition. As discussed in Section IV C, an attractive electron-phonon like interaction gives rise to both both a usual + +14 +s-wave τ0ψk state and an odd-parity τ3ψk state. These two states have the same pairing interaction, but the gap +projected onto the band basis is generally smaller for the τ3ψk state than for the τ0ψk state, implying that τ0ψk state +has the higher Tc. For the type 1 classes Dtype1 +2h,1 , Dtype1 +4h,1 , Dtype1 +4h,3 , and Dtype1 +4h,5 , anomalous pseudospin leads to Tc’s that +are nearly the same for the even τ0ψ and odd-parity τ3ψ states. These classes are therefore promising for observing +a field induced transition from an even-parity to an odd-parity state. +To determine if a such a field induced transition occurs we compute ˜Fk,ˆh for a pairing state ˜∆ = τ3. We find for +type 1 kp theories +˜Fk,ˆh = +(ˆh · λk)2(t2 +1,k + t2 +2,k + |λk|2) +|λk|2[ˆh2(t2 +1,k + t2 +2,k) + (ˆh · λk)2] +. +(19) +Notice if ˆh · λk = 0, then ˜Fk,ˆh = 0 which maximizes Tc. To determine the field orientations for which ˜Fk,ˆh = 0, we +examine the form of λk in the type 1 classes discussed above. In all these classes, the λz,k component appears with a +higher power of momenta than the other components. Consequently, the field should be applied along the ˆz direction. +As an example, consider the class Dtype1 +4h,3 . Here λz,k ∝ kxkykz(k2 +x − k2 +y) while λx,k ∝ ky and λy,k ∝ ky. In this case +λk will be in-plane to an excellent approximation, and an even to odd-parity transition can be expected for the field +along the c-axis. Consequently, classes Dtype1 +2h,1 , Dtype1 +4h,1 , Dtype1 +4h,3 , and Dtype1 +4h,5 +and, hence, space groups 56, 58, 59, 62, +128, 129, 130, 136, 137, and 138 are promising for realizing a field-induced even to odd parity transition. +2. +Field immune odd-parity superconductivity +For a conventional spin-triplet superconductor (with ∆ = dk · σ) formed from usual spin-1/2 pseudospin, SOC +typically pins the direction of the vector dk. If the applied field is perpendicular to dk, that is if dk · ˆh = 0, then the +Tc for this field orientation is unchanged [63–65]. Since there exists at least one field direction for which dk · ˆh ̸= 0, it +is not expected that usual spin-triplet superconductors are immune to fields applied in all directions. For anomalous +pseudopsin, this is not the case, it is possible for an odd-parity state to be robust against suppression for arbitrarily +oriented magnetic fields. To show how this is possible, we calculate ˜Fk,ˆh for ∆ = τ0(dk · σ) for type 1 kp theories, +this yields +˜Fk,ˆh = +[(t2 +1,k + t2 +2,k)dk · ˆh + (dk · λk)(λk · ˆh)]2 +[(t2 +1,k + t2 +2,k)ˆh2 + (λk · ˆh)2][(t2 +1,k + t2 +2,k)|dk|2 + (dk · λk)2] +. +(20) +We first note that near the nodal plane, the effective g-factor is small for in-plane fields ˆn·⃗h = 0, so that for these field +orientations superconductivity is not strongly suppressed (this is true for both even and odd-parity superconducting +states). Hence, to show that an odd-parity state survives for all field orientations, we need to show that ˜Fk,ˆh ≈ 0 +for a field applied along the nodal plane normal where λk · ˆh becomes maximal. Near the plane we expect that +λk · ˆh ≫ +� +t2 +1,k + t2 +2,k. Also, (t2 +1,k + t2 +2,k) is small compared to λ2 +k, so ˜Fk,ˆh is dominated by the dk · λk term in the +numerator. Hence if the denominator |t1,2dk| is much bigger than dk ·λk, then ˜Fk,ˆh ≈ 0. Given that λˆn is the largest +SOC component, this requirement is equivalent to λ⊥ ≪ t1,2 and dk ⊥ ˆn (where λ⊥ is the magnitude of the SOC +perpendicular to ˆn). +As a relevant example of the above mechanism we consider UPt3 [24]. The superconducting state in UPt3 is believed +to be an E2u state, with order parameter ∆ = ηp(σxky + σykx) + ηfσzkzkxky (we only include one component of +this two-component order parameter since similar arguments hold for the second component). In general, since the +p-wave and f-wave components have the same symmetry, both ηp and ηf are non-zero. However, theories based on +the usual pseudospin typically require ηp = 0 due to the experimental observations discussed below [66–68]. Below we +further show that ηp = 0 is not required for these experimental observations when anomalous pseudospin is considered. +Indeed, these experiments are consistent with ηf = 0 and ηp ̸= 0 if pairing occurs predominantly near the nodal plane +kz = π/c. +Thermal conductivity experiments suggest the existence of line nodes [24]. For usual pseudospin, the state σxky + +σykx is either fully gapped or has only point nodes. This is one reason to expect that ηp = 0. However, as illustrated +in Table II, line nodes are expected for this state on the kz = π/c plane (note this conclusion also follows from +Refs [18, 19, 21]). This is relevant for UPt3 since it is known to have the ‘starfish’ Fermi surface near this nodal plane +[24] which belongs to class Dtype1 +6h +In terms of paramagnetic suppression, the superconducting state is known to be more robust under B ⊥ z compared +to B ∥ z [68]. For the usual pseudospin, this requires dk ∥ z, and thus ηp = 0. However, on the ‘starfish’ Fermi + +15 +surface, the small g-factor for B ⊥ z can serve to protect the p-wave state against paramagnetic suppression. As +discussed above, the suppression from B ∥ z depends on the ratio λx,y/t1,2, while the g-factor for B ⊥ z depends +on the ratio (t1,2, λx,y)/λz. The requirement λx,y/t1,2 > (t1,2, λx,y)/λz is thus sufficient to match the observations +on the upper critical fields. If both ratios are much smaller than one, the p-wave state is immune to paramagnetic +suppression for field along arbitrary directions. This could be relevant to the approximately unchanged Knight shift +in the superconducting state [69]. We note that the use of ˜Fk,ˆh to determine the magnetic response relies on the +validity of projection to a single band. However, for class Dtype1 +6h +band degeneracies exist along three Dirac lines for +which this projection is not valid. In Appendix B we include a detailed numerical calculation that includes interband +effects. +VI. +8-FOLD DEGENERATE POINTS: APPLICATION TO UCOGE +The arguments presented above relied on the 4-fold degeneracy at TRIM points when SOC is not present. However, +some of these TRIM points have an 8-fold degeneracy without SOC. It is reasonable to ask if the conclusions found for +kp theories of 4-fold degenerate points discussed above survive to 8-fold degenerate points. To address this, we have +determined the symmetries of all orbital operators in Appendix C. We find that in most cases, the 8-fold degeneracy +at these TRIM is split by a single SOC term of the form Oσi where O is a momentum independent 4 by 4 orbital +matrix. In Table.IV, we give the direction of the spin component σi that appears in this SOC term at the TRIM point. +The existence of this single SOC term ensures small effective g-factors for fields perpendicular to the spin-component +direction. Consequently, the conclusions associated with the effective g-factor anisotropy discussed in Section V still +hold for these 8-fold degenerate points. We note that the 8-fold degeneracy at the A point of space groups 130 and +135 are not split by SOC and these points provide examples of double Dirac points examined in [70, 71]. +Spin Alignment +Space Group Momenta +σx +54(U1U2),54(R1R2),56(U1U2),60(R1R2),61(S1S2),62(S1S2),205(M1M2) +σy +52(S1S2),56(T1T2),57(T1T2),57(R1R2),61(T1T2),130(R1R2),138(R1R2) +σz +60(T1T2),60(U1U2),61(U1U2),62(R1R2),128(A3A4),137(A3A4),176(A2A3),193(A3),194(A3) +TABLE IV. Spin alignment of 8-fold degenerate TRIM. +One material for which these 8-fold degenerate points are likely to be relevant is the ferromagetic superconductor +UCoGe, which crystalizes in space group 62 (Pnma) [25]. UCoGe is believed to be a possibly topological odd-parity +superconductor [17, 25]. Our Fermi surface (given in Figure 3) reveals that all Fermi surface sheets lie near nodal +planes with anomalous pseudospin and further reveal tube-shaped pockets that enclose the zone-boundary S point +and stretch along the S-R axis. Here we focus on these Fermi surfaces. This feature reasonably agrees with previous +works [72–74] using local density approximation and the existence of these tube shaped Fermi surfaces is consistent +with quantum oscillation measurements [75]. Here density-functional theory calculations for UCoGe were carried +out by the full-potential linearized augmented plane wave method [56]. Perdew-Burke-Ernzerhof form of exchange +correlation functional [57], wave function and potential energy cutoffs of 16 and 200 Ry, respectively, muffin-tin sphere +radii of 1.4 ˚A for U and 1.2 ˚A for Co and Ge, respectively, the experimental lattice parameters [76], and an 8 × 12 × 8 +k-point mesh were employed for the self-consistent field calculation. Spin-orbit was fully taken into account in the +assumed nonmagnetic state. Fermi surface was determined on a dense 30 × 50 × 30 k-point mesh and visualized by +using FermiSurfer [77]. +Both the R and S points are 8-fold degenerate TRIM when SOC is not included for space group 62. Interestingly, +from Table IV, the effective g-factors for fields along ˆy and ˆz directions are zero at the S-point and are zero for fields +along ˆx and ˆy directions at the R-point. This indicates that superconductivity (both even and odd-parity) on the +tube-shaped Fermi surfaces will be robust against magnetic fields applied along the ˆy direction. This is the field +direction for which the upper critical field is observed to be the highest and for which an unusual S-shaped critical +field curve appears [25]. We leave a detailed examination of the consequences of anomalous pseudospin in space group +62 on superconductivity to a later work. +VII. +CONCLUSIONS +Non-symmorphic symmetries allow the existence of nodal planes at Brillouin zone edges when no SOC is present. +When SOC is added, the pseudospin on these nodal planes has different symmetry properties than usual pseudospin- + +16 +X +U +Z +Y +S +R +! +T +FIG. 5. +DFT Fermi surface of UCoGe. +1/2. Here we have classified all space groups and effective single-particle theories near TRIM points on these nodal +planes and examined the consequences of this anomalous pseudospin on the superconducting state. We have shown +how this enhances the Tc for odd-parity superconducting states due to attractive interactions, leads to unexpected +superconducting nodal properties, allows large Pauli limiting fields and pair density wave states for spin-singlet +superconductors, gives rise to field immune odd-parity superconductivity, and to field driven even to odd-parity +superconducting transitions. While we have emphasized nodal planes on which anomalous pseudospin exists, there +are also materials for which anomalous pseudospin develops on nodal lines and not on nodal planes. 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For the type 1 TRIM, we write the gap +functions in terms of the complete set of gap functions with the correct symmetries given in Table III as +∆1,++ = +ψ0τ0 + (dz · ˆn)(σ · ˆn)τ3 +∆1,+− = +ψxτ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2 +∆1,−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2 +∆1,−− = +(d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (ψ · ˆn)(σ · ˆn)τ2 +(A3) +where di are odd functions of k and ψi are even functions of k. +Using Eq. A1, the corresponding quasiparticle +excitation energies can be found to be +E1,++ = +±′ � +(ϵ0 ± λ · ˆn)2 + (ψ0 ± dz · ˆn)2 +E1,+− = +±′ �� +ϵ2 +0 + ψ2x + (dz × ˆn)2 + d2y ± λ · ˆn +� +E1,−+ = ±′ � +(ϵ0 ± λ · ˆn)2 + (ψz ± d0 · ˆn)2 + (dx × ˆn)2 + (ψ × ˆn)2 ± 2(dx × ψ) · ˆn +E1,−− = +±′ �� +ϵ2 +0 + (d0 × ˆn)2 + (dx · ˆn)2 + (ψ · ˆn)2 ± λ · ˆn +� +(A4) +where the prime denotes independent choices of the sign. For type 2 TRIM we similarly have +∆2,++ = +ψ0τ0 + (ψ · ˆn)(σ · ˆn)τ2 +∆2,+− = +ψxτ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2 +∆2,−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2 +∆2,−− = +(d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (dz · ˆn)(σ · ˆn)τ3 +(A5) +The quasiparticle excitation spectra for these states are +E2,++ = +±′ � +(ϵ0 ± λ · ˆn)2 + (ψ0 ± ψ · ˆn)2 +E2,+− = +±′ �� +ϵ2 +0 + ψ2x + ψ2z + (ψ × ˆn)2 ± λ · ˆn +� +E2,−+ = ±′ � +(ϵ0 ± λ · ˆn)2 + (dy ± d0 · ˆn)2 + (dx × ˆn)2 + (dz × ˆn)2 ± 2(dx × dz) · ˆn +E2,−− = +±′ �� +ϵ2 +0 + (d0 × ˆn)2 + (dx · ˆn)2 + (dz · ˆn)2 ± λ · ˆn +� +(A6) + +21 +Appendix B: Magnetic susceptibility UPt3 +In the main text, we illustrated how the p-wave state in UPt3 is immune to the magnetic field along arbitrary +directions. +An important step is to consider the small g-factor for field B ⊥ z. +However, the discussion is not +complete. In the normal state, there exist 4-fold degenerate Dirac lines on the plane kz = π/c, where the g-factor is +not small. In terms of the field fitness, Eq.20 in the main text only considered doubly degenerate bands. In principle, +extra terms in the field fitness are needed for to describe these Dirac lines. However, the Fermi surface is not right +on the nodal plane. This can make the Dirac lines unimportant. In this section, we will explicitly check the field +response in the superconducting state through a numerical calculation on a tight-binding model for UPt3. +In the following calculations, we will focus on the Knight shift (spin-susceptibility). Knight shift measures spin +polarization at atom sites. By extracting spin susceptibility χs, one can determine pairing functions of an uncon- +ventional superconductor. For a single-band spin-triplet superconductor, the change of Knight shift depends on the +orientation of magnetic field with respect to the d-vector of the superconducting state. If the magnetic field is per- +pendicular to the d-vector, the Knight shift should be a constant across superconducting Tc. If the magnetic field is +parallel to the d-vector, the Knight shift will decrease to zero as temperature approaches zero. For the multi-band +non-symmorphic superconductor UPt3, Knight shift is almost unchanged for all field orientations, suggesting the +importance of spin-orbit coupling in this heavy fermion material. +One of the Fermi surfaces (‘starfish’) of UPt3 is flat and located near the high symmetry plane kz = π/c. Zeeman +terms Bxσx and Byσy then becomes inter-band. From non-generate perturbation theory, spin susceptibilities are +inversely proportional to the band gap. This is different from the intra-band Zeeman effect, where susceptibilities are +proportional to the density of states on Fermi surface, according to degenerate perturbation theory. +Since the superconducting gap is much smaller than the band gap, inter-band susceptibilities will be unchanged +across Tc. If the superconductivity is mainly developed on the above flat Fermi surface, then Knight shift is expected +to be unchanged for in-plane magnetic fields, regardless of the superconducting pairing symmetry. If the d-vector is +in-plane, then Knight shift will also be unchanged for a perpendicular magnetic field. In this section, we will explicitly +illustrate this idea to understand the experimental results on UPt3. +FIG. 6. Crystal structure of UPt3 with the unit vector e1 = (1, 0, 0). +The 4 × 4 normal state Hamiltonian reads [67]: +H = ε(k) + gz(k)σzτ3 + a1(k)τ1 + a2(k)τ2 + [gx(k)σx + gy(k)σy] τ3 +εk = 2t +� +i=1,2,3 +cos k∥ · ei + 2t3 cos kz − µ, +gz(k) = gz0 +� +i +sin k∥ · ei +a1(k) = 2t′ sin kz +2 +� +i=1,2,3 +sin k∥ · ri, +a2(k) = 2t′ sin kz +2 +� +i=1,2,3 +cos k∥ · ri +gx(k) = gx0fxfy sin kz, +gy(k) = gy0(f 2 +x − f 2 +y ) sin kz +fx ≡ sin k∥ · e1 − sin k∥ · e2 + sin k∥ · e3 +2 +, +fy ≡ +√ +3 sin k∥ · e2 − sin k∥ · e3, +(B1) +here (kx, ky, kz) are relative to the high symmetry point (0, 0, π). Relevant vectors ei and ri can be found in Fig.6. +τi matrices live in the sublattice space. On the high-symmetry plane kz = 0, the inter-sublattice hopping a1,2 and +the spin-flip SOC gx,y vanish. |k, m = 1, ↑⟩ and |k, m = 2, ↓⟩ states form a pseudospin band, while |k, m = 2, ↑⟩ and +|k, m = 1, ↓⟩ states form another band. + +22 +We now study spin susceptibilities. We will focus on a p-wave state in the E2u channel. Its d-vector is in-plane: +d = ∆(T)(fx, −fy, 0). fx and fy are introduced in Eq.B1, and they transform as kx and ky. The gap magnitude is +taken to be ∆(T) = ∆0 +� +1 − T/Tc. t = 1, t3 = −4, gz0 = 2, µ = 12 and ∆0 = Tc = 0.001 is taken in the calculation. +0 +0.2 +0.4 +0.6 +0.8 +1 +T/Tc +0 +0.02 +0.04 +0.06 +0.08 +0.1 +t'=0, gx0=gy0=0 +x +z +0 +0.2 +0.4 +0.6 +0.8 +1 +T/Tc +0 +0.02 +0.04 +0.06 +0.08 +0.1 +t'=0, gx0=gy0=0.1 +x +z +x,inter +z,inter +0 +0.2 +0.4 +0.6 +0.8 +1 +T/Tc +0 +0.02 +0.04 +0.06 +0.08 +0.1 +t'=0.1, gx0=gy0=0 +x +z +x,inter +z,inter +FIG. 7. Spin susceptibilities as a function of temperature, for (left) a1 = a2 = gx = gy = 0, which would be the case if the +Fermi surface exactly lied on the high-symmetry plane. (middle) non-zero spin-flip SOC but zero inter-sublattice hopping. +(right) non-zero inter-sublattice hopping but zero spin-flip SOC. +To illustrate the effect of the anomalous pseudospin, we start with a toy model with zero inter-sublattice hopping +and spin-flip SOC: t′ = gx0 = gy0. The corresponding four terms vanish in the normal state Hamiltonian: a1 = a2 = +gx = gy = 0. In this extreme case, the spin susceptibilities are unchanged across Tc, as shown in the left panel of +Fig.7. +We now turn on the spin-flip SOC (gx0 and gy0), while keeping the inter-sublattice hopping t′ to be zero. hxσx +develops an intra-band component, which will be suppressed in the superconducting state. As a result, the total +χx deep in the superconducting state starts to decrease as function of temperature. For χz, spin-flip SOC induces +higher-order terms in the E2u channel. The d-vector develops non-zero z-component in the band basis. This causes +a decrease in χz. The result for gx0 = gy0 can be found in the middle panel of Fig.7. The inter-band susceptibilities +in the normal state are included in dashed lines. +We now turn on the inter-sublattice hopping t′, while keeping the spin-flip SOC (gx0 and gy0) to be zero. A similar +effect is expected for χx due to the intra-band contribution. For χz, since σz is a good quantum number, χz will be +unchanged. The result can be found in the right panel of Fig.7. +Experimentally, the superconducting state is known to be more robust under B ∥ x compared to B ∥ z. In other +words, the decrease in χx needs to be smaller than χz. This scenario is closer to the second limit. + +23 +Appendix C: 8-fold Representations +Here, we list the symmetries of all orbital operators near the 8-fold degenerate points. The point group that keeps +the TRIM point invariant can be found in the title. The bracket notation [·] is also used for antisymmetric operators +which was τ2 in the main context, but in 8-fold cases, the antisymmtric component is not unique due to the higher +degrees of freedom. +Space group momenta +Point group D2h +54(U1U2) +Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] +54(R1R2) +Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] +56(U1U2) +Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] +60(R1R2) +Ag + 2B1g + 2B2g + B3g + Au + 2B2u + B3u + [Ag] + [B3g] + [Au] + 2[B1u] + [B3u] +61(S1S2) +Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] +62(S1S2) +Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] +205(M1M2) +Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] +52(S1S2) +Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] +56(T1T2) +Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] +57(T1T2) +Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] +57(R1R2) +Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] +61(T1T2) +Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] +130(R1R2) +Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] +138(R1R2) +Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] +60(T1T2) +Ag + B1g + 2B2g + 2B3g + 2Au + B2u + B3u + [Ag] + [B1g] + 2[B1u] + [B2u] + [B3u] +60(U1U2) +Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u] +61(U1U2) +Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u] +62(R1R2) +Ag + B1g + 2B2g + 2B3g + 2B1u + B2u + B3u + [Ag] + [B1g] + 2[Au] + [B2u] + [B3u] +Space group momenta +Point group D4h +128(A3A4) +A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u] +137(A3A4) +A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u] +Space group momenta +Point group C6h +176(A2A3) +Ag + Bg + E1g + E2g + Au + Bu + E1u + [Ag] + [Bg] + [Au] + [Bu] + [E2u] +Space group momenta +Point group D6h +193(A3) +A1g + B2g + E1g + E2g + A1u + B1u + E1u + [A2g] + [B1g] + [A2u] + [B2u] + [E2u] +194(A3) +A1g + B1g + E1g + E2g + A1u + B2u + E1u + [A2g] + [B2g] + [A2u] + [B1u] + [E2u] +TABLE V. Symmetries of orbital operators at the 8-fold degenerate points. + diff --git a/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/load_file.txt b/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82a0d944a4284010761997367f83ea546a0e5183 --- /dev/null +++ b/PtFJT4oBgHgl3EQfJCwC/content/tmp_files/load_file.txt @@ -0,0 +1,1928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf,len=1927 +page_content='Superconductivity of anomalous pseudospin Han Gyeol Suh1, Yue Yu1,2, Tatsuya Shishidou1, Michael Weinert1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Brydon3, and Daniel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Agterberg1 1Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53201, USA 2Department of Physics, Stanford University, 476 Lomita Mall, Stanford, CA 94305, USA and 3Department of Physics and MacDiarmid Institute for Advanced Materials and Nanotechnology, University of Otago, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Box 56, Dunedin 9054, New Zealand In materials with both time-reversal (T) and inversion symmetry (I), superconductivity is formed by pairing fermion pseudospin partners at momenta k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Typically, pseudospin shares the same symmetry properties as usual spin-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we consider non-symmorphic materials with mo- mentum space spin-textures that exhibit an anomalous pseudospin with different symmetry prop- erties than usual spin-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We provide a comprehensive list of space groups for which anomalous pseudospin occurs on planes in momentum space and carry out a complete categorization and anal- ysis of superconductivity for Fermi surfaces centered on all possible T, I invariant momenta (TRIM) in these planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We show that superconductivity from this anomalous pseudospin leads to a vari- ety of unusual consequences for superconductivity including: extremely large Pauli limiting fields and residual Knight shifts for pseudospin singlet superconductors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' field induced pair density wave states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' field induced pseudospin singlet to pseudospin triplet transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' fully gapped ‘nodal’ super- conductors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' and additional insight into the breakdown of Blount’s theorem for pseudospin triplet superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We apply our results to UPt3, BiS2-based superconductors, Fe-based supercon- ductors, and paramagnetic UCoGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' INTRODUCTION Momentum space spin-textures of electronic bands are known to underlie spintronic and superconducting properties of quantum materials [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the spintronics context, Rashba-like spin textures allow control of electronic spin through applied electric fields [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In superconductors, these same spin textures lead to unusual and counter-intuitive magnetic response, such as the robustness of spin-singlet superconductivity to applied magnetic fields, pair density wave states, and singlet-triplet mixing [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' While such spin-textures are common when inversion symmetry is broken, it has been realized that these can occur when inversion symmetry is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This has lead to the notion of hidden spin-textures [4] and locally non-centrosymmetric superconductivity [5], where inversion related sectors each allow a Rashba-like spin-texture due to the local inversion symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These spin-textures are of opposite sign on the two sectors, so that global inversion symmetry is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These hidden spin-textures allows the novel physics associated with spin-orbit coupling (SOC) to emerge even when inversion symmetry is not broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It further allows for new physics to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' One notable example is a field induced transition from an even-parity (pseudospin singlet) to odd-parity (pseudospin triplet) observed in CeRh2As2 [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Key to observing novel physics associated with these spin-textures in inversion symmetric materials, is that the inversion related sectors are weakly coupled [5, 9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Theoretical proposals for how to achieve this fall under two approaches: the first is to tailor weak coupling between the inversion related sectors, for example by separating two inversion symmetry related layers so that the interlayer coupling is weak [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' the second is to exploit symmetries that ensure that this inter-sector coupling vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The symmetry based approach has been applied to points and lines in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Examples include two-dimensional (2D) transition metal dichalcogenides near the K-point [12] and non-symmorphic symmetries near the X −M line in BaNiS2 with space group 129 (P4/nmm) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In these cases, the only energy splitting between the inversion-related sectors is due to SOC - a situation conceptually similar to materials with broken inversion symmetry, where the usual two-fold pseudopsin degeneracy is broken solely by SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we generalize this symmetry based approach by identifying electronic band degeneracies that are split solely by SOC in materials with both inversion, I, and time-reversal, T, symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This requires bands that are at least four-fold degenerate when SOC is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Such band degeneracies are not generic and require symmetries beyond the usual two-fold pseudospin (or Kramers) degeneracy that arises from TI symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we focus on 2D momentum planes, nodal planes, where this occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is the largest region in momentum space for which the required four- fold electronic degeneracies can appear when SOC is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed in a variety of contexts [13–16], such nodal planes arise in non-symmorphic crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we provide a complete list of space groups for which this occurs and provide symmetry based kp theories for all time-reversal-invariant momenta (TRIM) on these nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed later, many relevant superconductors exhibit Fermi surfaces near these TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We find that the SOC-split electronic states on these nodal planes generically exhibit a pseudospin that has a different symmetry than that of usual spin-1/2 fermions (this generalizes a result we found for space group P4/nmm [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we name this anomalous pseudospin and examine the consequences of this anomalous pseudospin on superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We find that this anomalous pseudospin plays a central role on the superconducting magnetic response and on the properties arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='11458v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='supr-con] 26 Jan 2023 2 of spin-triplet superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Our results complement and provide further insight on earlier nodal and topological classifications of superconductivity in non-symmorphic materials [17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this paper we begin by defining anomalous pseudospin on nodal momenta planes, we then characterize all possible symmetry based kp theories near TRIM points on these nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Using these kp theories, we analyse the magnetic response and nodal excitations of superconducting states formed from anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We apply this analysis to a series of materials that exhibit Fermi surfaces that lie on or near these nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' More specifically we reveal how anomalous pseudospin: explains critical fields that far exceed the Pauli field in BiS2-based materials [22] and the observed magnetic response 3D Fe-based superconductors [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' identifies which space groups and TRIM are ideal to find a field induced even parity to odd parity transition akin to that observed in CeRh2As2 [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' provides insight into the gap symmetry of UPt3 [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' and shines new light on re-entrant superconductivity in UCoGe [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' ANOMALOUS PSEUDOSPIN: SYMMETRY ORIGIN Our aim is to exploit symmetry to find nodal plane band degeneracies that are lifted solely by SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed below, once these band degeneracies are lifted, a two-fold pseudospin degeneracy will remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We find that generically, the pseudospin that results from this procedure does not share the same symmetry properties as usual spin 1/2 and hence we name this anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Pseudospin describes the two-fold Kramers degeneracy that arises at each momentum point k when the product of time-reversal T and inversion I symmetries, TI, is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The product TI is anti-unitary and for fermions satisfies (TI)2 = −1, ensuring at least a two-fold degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is often the case that this pseudospin behaves as spin-1/2 under rotations [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, when symmetries beyond TI are present, it is possible that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' One example of this is the angular momentum jz = ±3/2 electronic states that arise when cubic symmetry or a three-fold rotation axis is present [2, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the latter case, this gives rise to so-called type-II Ising superconductivity in 2D materials [28, 29] where large in-plane critical fields appear when the Fermi surface is sufficiently close to momentum points with this three-fold rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In our case, the anomalous pseudospin appears on momentum planes in the Brillouin zone, allowing a larger phase space for the physical properties of anomalous pseudospin to manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To ensure the requisite band degeneracy on a nodal plane, consider the symmetry elements that keep a momentum point on the plane invariant (here taken to be normal to the ˆn axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These are {E, ˜ Mˆn, TI, T ˜C2,ˆn}, where ˜ Mˆn is a translation mirror symmetry and ˜C2,ˆn is a translation two-fold rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Their point group rotation and translation component can be denoted using Seitz notation, for example ˜ Mˆn = {Mˆn|t1, t2, t3} where Mˆn is a point group mirror symmetry along ˆn and (t1, t2, t3) is a fractional translation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since we are searching for a degeneracy that appears without SOC, we consider orbital or sublattice degrees of freedom for which (TI)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The only remaining symmetry that can enforce a two-fold degeneracy is T ˜C2,ˆn, since this is anti-unitary, it must satisfy (T ˜C2,ˆn)2 = −1 to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since T commutes with rotations, this implies ˜C2 2,ˆn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When operating on orbital or sublattice degrees of freedom, ˜C2 2,ˆn is typically 1, suggesting it is not possible to have the required degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, in non-symmorphic groups, ˜C2,ˆn can be a screw axis, for which it is possible to satisfy ˜C2 2,ˆn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, using Seitz notation ˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (here t1 and t2 correspond to either a half in-plane translation vector or to no translation) we have ( ˜C2,ˆn)2 = {E|0, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When operating on a state carrying momentum k, ( ˜C2,ˆn)2 is represented by eik·ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Hence if the nodal plane sits at momentum k · ˆn = π, then ˜C2 2,ˆn = −1 and a two-fold orbital or sublattice degeneracy is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When spin-degeneracy is also included, these states are then four-fold degenerate when SOC is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When SOC is included, it is possible to show that the TI pseudospin partners have the same Mˆn mirror eigenvalue (this result is generalization of that given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' [9] where t1 = 0 and t2 = 0 was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' That is, labeling the two Kramers degenerate states as |+⟩ and TI|+⟩, both belong to the same eigenstate of ˜ Mˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As a consequence, all Pauli matrices ˜σi made from the two states |+⟩ TI|+⟩ must all be invariant under ˜ Mˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is this feature that differs from usual spin-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Of the three Pauli matrices σi, constructed from usual spin-1/2 states, two will be odd under ˜ Mˆn and one will be even under ˜ Mˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is this symmetry distinction between the anomalous pseudospin operators (˜σi) and usual spin 1/2 operators (σi) that underlie the unusual superconducting properties discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The above argument can also be applied to nodal lines generated by the symmetry elements {E, ˜C2,ˆn, TI, T ˜ Mˆn} with (T ˜ Mˆn)2 = −1 when applied to orbital or sublattice degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this case, repeating the same arguments above show that SOC will also split the band degeneracy and lead to anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here, due to the larger available momentum phase space, we restrict our analysis and classification to nodal planes and leave an analysis of nodal lines to a later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For all space groups that host nodal planes, we develop symmetry-based kp theories valid near all TRIM on theses nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We emphasis these TRIM since Cooper pairs are formed by pairing states at momenta k and −k with the momentum origin given by a TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We then consider Fermi surfaces 3 Z E B D Y C A Γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Example from space group 14 where the green shading reveals the planes and lines in momentum space on which anomalous pseudospin exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A Fermi surface located near the momentum plane kz = π (as depicted by the dark Fermi surface near the Z point) will have its superconducting properties governed by pairing of anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, Fermi surfaces far from these planes (such as that depicted near the Γ point) will exhibit more usual superconducting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' near these TRIM and discuss the resultant superconducting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Figure 1 illustrates our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here, in green, we show the nodal planes and lines that exhibit anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we examine the properties of superconductivity for a Fermi surface near the Z point, which is a TRIM on the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The properties of superconductivity for a Fermi surface near the Γ point, for which pseudospin is typically not anomalous, are described in earlier review articles [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that many superconducting materials, including the examples discussed in this paper, exhibit Fermi surfaces near nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' NODAL PLANE SPACE GROUPS AND SINGLE-PARTICLE kp HAMILTONIANS Here we identify all space groups that allow anomalous pseudospin on nodal planes and construct the corresponding symmetry-based kp-like Hamiltonians for all TRIM on these planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Space groups with nodal planes To identify these nodal planes, all space groups containing inversion symmetry I = {I|0, 0, 0} and the screw axis ˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (where t1 = 0, 1/2 and t2 = 0, 1/2) were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For these space groups, the nodal planes lie on the Brillouin zone boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Table 1 lists the resultant space groups, point groups, nodal planes, and types of kp theories allowed for these space groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed in the previous section, the degeneracies of these nodal planes is generically lifted by SOC, yielding anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Symmetry based kp theories near TRIM To understand the consequences of anomalous pseudospin on superconductivity requires a theory for the normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Cooper pairs rely on the degeneracy between states of momenta k and −k and this degeneracy is ensured by both T and I symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For this reason, we develop symmetry-based kp theories expanded around TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To derive these kp-like Hamiltonians,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' we have used the real representations for the TRIM given in the Bilbao Crystallographic 4 Crystal Type Number Name Nodal planes kp theory classes Monoclinic (C2h) 11 P21/m (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Ctype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 14 P21/c (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Ctype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Ctype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 Orthorhombic (D2h) 51 Pmma (1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 52 Pnna (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold 53 Pmna (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 54 Pcca (1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold 55 Pbam (1/2,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 56 Pccn (1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 ,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype2 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 137 P42/nmc (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold 138 P42/ncm (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype2 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype2 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold Hexagonal (C6h) 176 P63/m (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2) Ctype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Ctype1 6h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold Hexagonal (D6h) 193 P63/mcm (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 6h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold 194 P63/mmc (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Dtype1 6h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold Cubic (Th) 205 Pa3 (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' w) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-fold TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Space groups with nodal planes server [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For these TRIM, we initially consider space group irreducible representations that do not include spin, which, for simplicity, we name orbital representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These representations are either 2-fold or 4-fold degenerate (when spin is added, these becomes 4-fold and 8-fold degenerate respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The full kp-like Hamiltonians are only listed for the 2-fold degenerate representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We present a partial classification of the 4-fold degenerate orbital representations near the end of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In constructing the kp theories for the 2-fold orbital degenerate TRIM points, we choose τi to be Pauli matrices that encode the orbital degrees of freedom, and σi to be spin Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We take T = τ0(iσy)K where K is the complex conjugation operator, hence the τ2 operator is odd under time-reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For a given doubly degenerate space group representation on a TRIM, constructing its direct product leads to four irreducible point group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These four representations each correspond to an orbital operator τi, and this partially dictates the momentum dependencies of symmetry allowed terms in the kp Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We present our results for the kp Hamiltonians in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first row of each box gives the type of the kp theory class and the point group representations of the orbital operators that are given by Pauli matrices τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this decomposition, the square brackets correspond to the antisymmetric τ2 operator and remaining terms correspond to τ0, τ1, and τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The second row of a box gives the kp Hamiltonian, and the last part of a box lists the space groups and TRIM points representations that belong to the kp Hamiltonian class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We have tabulated the kp Hamiltonians for 122 TRIM points and we find that only 13 different kp theories appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These are of two types, which we call type 1 and type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Type 1 kp theories have degenerate even and odd parity orbital basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Type 2 kp theories has two degenerate orbital basis functions with the same parity symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The generic form of these kp theories are H(k) = ε0,k + t1,kτ1 + tα,kτα + τβ(λk · σ) = ε0,k + Hδ(k) , (1) 5 (I, τα, τβ) = � (τ1, τ2, τ3) for type 1 , (τ0, τ3, τ2) for type 2 , (2) where Hδ(k) = H(k) − ε0,k and α and β are type indices will be used the remaining context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For parity mixed, type 1, kp theories, the degeneracy at TRIM points is not broken by SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is because the non-symmorphic symmetries combined with topological arguments imply these TRIM must have an odd number of Dirac lines passing through them [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These Dirac lines lie in the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Elsewhere in the nodal plane, SOC lifts the 4-fold degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We will discuss some consequences of these Dirac lines later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The non trivial inversion symmetry for type 1, I = τ1, implies the parity of the momentum functions that ε0,k = ε0,−k, t1,k = t1,−k, t2,k = −t2,−k, and λk = −λ−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This form of Hamiltonian has often been used to understand locally non-centrosymmetric superconductors [2] and hidden spin polarization in inversion symmetric materials [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In these contexts, the orbital degrees of freedom reside on different sectors that are related by inversion symmetry and there is typically no symmetry requirement that ensures the SOC dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The τ3 matrix is odd under inversion symmetry, allowing the odd-parity SOC λk to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Many superconductors of interest have Fermi surfaces near type 1 TRIM points, examples include: Fe-based superconductors, which often have electron pockets near the M point in space group 129 (classes Dtype1 4h,1 or Dtype1 4h,3 ) [23], in this context the high Tc superconductor monolayer FeSe is of interest, since it only has Fermi surfaces near the M point [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' CeRh2As2 which exhibits a field induced transition from an even parity to an odd-parity superconducting state [7, 8] and has Fermi surfaces near the M point in space group 129 (classes Dtype1 4h,1 or Dtype1 4h,3 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' BiS2-based superconductors [22] which has superconductivity that survives to very high fields and which has electron pockets near the X point in space group 129 (class Dtype1 2h,3 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' the odd-parity heavy fermion superconductor UPt3 [24] which has a pancake-like Fermi surface at kz = π/c in space group 193 (class Dtype1 6h );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' and the ferromagnetic superconductor UCoGe [25] with space group 62 and a Fermi surface near the T point (class Dtype1 2h,1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For type 2 kp theories, the 4-fold degeneracy is sometimes already split into 2 at the TRIM point when SOC is added, unlike what occurs for type 1 kp theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This happens in classes Ctype2 2h,2 and Dtype2 2h,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the other type 2 classes, this degeneracy at the TRIM point is not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In these cases, an even number of Dirac lines pass through the TRIM point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These Dirac lines lie in the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since I = τ0 for type 2, all terms in the Hamiltonian are even parity, that is, unchanged under k → −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' One example where type 2 kp theories apply is in strain induced superconductivity in RuO2[37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Without strain, RuO2 is thought to be a non-superconducting altermagnet [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When strain is applied, bands near the X-M-R-A Brillouin zone face are most strongly affected [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' RuO2 has space group 136 with the R and M points belonging to classes Dtype2 2h,4 , Dtype2 4h,2 , or Dtype2 4h,4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Later we discuss the ferromagnetic superconductor UCoGe with space group 62 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this example we highlight the role of 8-fold degenerate points which exhibit some properties similar to that found for type 2 TRIM points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Type 1 and type 2 kp Hamiltonians share some common features that play an important role in understanding the properties of the superconducting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first is that the non-symmorphic symmetry dictates that these Hamiltonians are best described as two-band systems with eigenenergies given by E±(k) = ε0,k ± � t2 1,k + t2 α,k + |λk|2 = ε0,k ± εδ,k , (3) where α is the type index in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The second feature is that both simplify dramatically on the nodal plane, where only the coefficient functions ε0,k and λk·ˆn are non-vanishing (that is t1,k = t2,k = t3,k = |λk׈n| = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This property is a direct consequence of the anomalous pseudopspin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The symmetry arguments discussed in the previous section enforce this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, for momenta on the nodal plane, the mirror operator through the nodal plane, UM, takes the from UM = −iτβ(σ · ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The requirement that these Hamiltonians obey time-reversal and inversion symmetries and commute with UM lead to this simple form of the kp theories in the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The final important property of these kp Hamiltonians is that the SOC terms are often the leading order terms in the kp expansions, that is, they appear with the lowest powers of ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is the case for classes Ctype2 2h,2 , Dtype1 2h,1 , Dtype2 2h,4 , Dtype1 4h,2 , Dtype1 4h,3 , and Dtype1 4h,5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This feature ensures that there exists a limit in which the SOC is the dominant single-particle interaction on the Fermi surface and hence the unusual magnetic superconducting response we later discuss must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' SUPERCONDUCTING STATES In the previous section, complete symmetry-dictated kp theories were found for anomalous pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These theories are complete in the sense that they include all operators of the form τiσj allowed by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For super- conductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' the orbital degree of freedom enlarges the corresponding space of possible gap functions compared to the usual even-parity (pseudospin-singlet) ˜∆(k) = ψk(iσy) and odd-parity (pseudospin-triplet) ˜∆(k) = dk · σ(iσy) 6 Class Symmetry Hamiltonian Space Group Momenta Ctype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 Ag + Bg + [Au] + Bu H = ϵ0 + (t1xkx + t1zkz)kyτ1 + t2kyτ2 + τ3[λxkyσx + (λyxkx + λyzkz)σy + λzkyσz] 11(C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Z1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 14(Z1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 63(R1(yz)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 176(L1(yz)) Ctype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 Ag + 2Bg + [Ag] H = ϵ0 + (t1xkx + t1zkz)kyτ1 + (t3xkx + t3zkz)kyτ3 +τ2[(λxxkx + λxzkz)kyσx + λyσy + (λzxkx + λzzkz)kyσz] 14(D± 1 D± 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 64(R± 1 R± 2 (yz)) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 Ag + B1g + [Au] + B1u H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2 + τ3[λxkyσx + λykxσy + λzkxkykzσz] 56(S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 58(R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2) 59(S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 62(T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xz)) Dtype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 Ag + 2B1g + [Ag] H = ϵ0 + t1kxkyτ1 + t3kxkyτ3 + τ2[λxkykzσx + λykxkzσy + λzkxkyσz] 55(S± 1 S± 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' S± 3 S± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R± 1 R± 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R± 3 R± 4 ) 56(R± 1 R± 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R± 3 R± 4 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 58(S± 1 S± 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' S± 3 S± 4 ) Dtype1 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 Ag + B2g + [B1u] + B3u H = ϵ0 + t1kxkzτ1 + t2kzτ2 + τ3[λxkxkykzσx + λykzσy + λzkyσz] 51(X1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(zyx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(zyx)) 127(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 128(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz)) 129(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 130(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)) 135(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 136(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz)) 137(R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 138(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)) 193(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 194(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)) 205(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xyz)) Dtype2 2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 Ag + B1g + B3g + [B2g] H = ϵ0 + t1kxkyτ1 + t3kykzτ3 + τ2[λxkxkyσx + λyσy + λzkykzσz] 52(T ± 1 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 53(U ± 1 (yz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' R± 1 (yz)) 58(T ± 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' U ± 1 (xy)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 60(S± 1 (xy)) 128(R± 1 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 136(R± 1 ) Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 A1g + B2g + [A1u] + B2u H = ϵ0 + t1kxkyτ1 + t2kxkykz(k2 x − k2 y)τ2 + τ3[λx(kxσy + kyσx) + λ3kxkykzσz] 129(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 130(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2) 136(A3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 137(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 138(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2) Dtype2 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 A1g + 2B2g + [A1g] H = ϵ0 + t1kxkyτ1 + t3kxkyτ3 + τ2[λx(kykzσx + kxkzσy) + λzkxky(k2 x − k2 y)σz] 127(M ± 1 M ± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' M ± 2 M ± 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A± 1 A± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A± 2 A± 3 ) 128(M ± 1 M ± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' M ± 2 M ± 3 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 135(M ± 1 M ± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' M ± 2 M ± 3 ) 136(M ± 1 M ± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' M ± 2 M ± 3 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 138(A± 1 A± 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A± 2 A± 3 ) Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 A1g + B2g + [B1u] + A2u H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2 + τ3[λx(kxσy − kyσx) + λzkxkykz(k2 x − k2 y)σz] 129(M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 130(M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4) 136(A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 137(M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 138(M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4) Dtype2 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 A1g + A2g + B2g + [B1g] H = ϵ0 + t1kxky(k2 x − k2 y)τ1 + t3kxkyτ3 + τ2[λx(kykzσx + kxkzσy) + λzkxkyσz] 127(M ± 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A± 5 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 128(M ± 5 ) 135(M ± 5 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 136(M ± 5 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 138(A± 5 ) Dtype1 4h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 A1g + A2g + [B1u] + B2u H = ϵ0 + t1kxky(k2 x − k2 y)τ1 + t2kxkykzτ2 + τ3[λx(kxσy + kyσx) + λzkxkykzσz] 128(A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 137(A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2) Ctype1 6h Ag + Bg + [Au] + Bu H = ϵ0 + (t1xkx(k2 x − 3k2 y) + t1yky(3k2 x − k2 y))kzτ1 + t2kzτ2 + τ3[λxkz(2kxkyσx + (k2 x − k2 y)σy) + (λzxkx(k2 x − 3k2 y) + λzyky(3k2 x − k2 y))σz] 176(A1) Dtype1 6h A1g + B2g + [A2u] + B1u H = ϵ0 + t1kxkz(k2 x − 3k2 y)τ1 + t2kzτ2 +τ3[λxkz(2kxkyσx + (k2 x − k2 y)σy) + λzky(3k2 x − k2 y)σz] 193(A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 194(A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2(xy)) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Classification of kp theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Subscript numbering of momenta represents different real representations on the same momentum point, and a permutation of the axes is denoted by the cyclic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For example, 128(X1,2(xyz)) represents that there are two representations X1 and X2 on X = (0, 1/2, 0) space group 128, and their local theory is obtained by Dtype1 2h,3 Hamiltonian under x → y → z → x relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The representation convention is following Bilbao Crystallographic servera[32–34] except for the L point in 193 and 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' a https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='ehu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='es/ Representations and Applications → Point and Space Groups → - Representations → SG Physically irreducible representations given in a real basis 7 states that appear in single-band theories [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Nevertheless, it is possible to understand some general properties of the allowed pairing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To deduce the symmetry properties of possible pairing channels in this larger space of electronic states, it is useful to define gap function differently than usual [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, we take H = � i,j,k Hij(k)c† k,ick,j + 1 2 � i,j,k [∆ij(k)c† k,i˜c† k,j + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (4) where i, j are combined spin and orbital indices, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' means Hermitian conjugate, ck(c† k) is the Fermionic spin-half particle creation(annihilation) operator, and ˜ck(˜c† k) is the time reversed partner of ck(c† k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the usual formulation ˜c† k,j is replaced c† −k,j which leads a different gap function ˜∆ij and to difficulties in interpreting the symmetry transformation properties of this gap function [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For a single-band, these new gap functions become ∆(k) = ψkσ0 for even- parity and ∆(k) = dk · σ for odd-parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The key use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 4 is that the ∆ij(k) transform under rotations in the same way as the Hij(k), allowing the symmetry properties of the gap functions to be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The disadvantage of this approach is that the antisymmetry of the gap functions that follows from the Pauli exclusion principle is not as readily apparent compared to the usual formulation [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Enforcing the Pauli exclusion principle leads to eight types of gap functions that generalize the pseudospin-singlet and pseudospin-triplet of single-band gap functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Six of these are simple generalizations of the single-band gap functions: τiψk and τi(dk · σ) for i = 0, 1, and 3 where ψ−k = ψk and d−k = −dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Two are new gap functions: τ2(ψk · σ) and τ2dk with ψ−k = ψk and d−k = −dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is possible to determine whether these gaps functions are either even or odd-parity and this depends upon whether the kp Hamiltonian is type 1 or type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These gap functions and their parity symmetry are listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Without further consideration of additional symmetries, the gap function will in general be a linear combination all the even (or odd) parity gap functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To gain an understanding of the relative importance of these pairing states it is useful to project these gaps onto the band basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Such a projection is meaningful if the energy separation between the two bands is much larger than the gap magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For many of the kp Hamiltonians, due to the presence of Dirac lines, there will exist regions in momentum space for which this condition is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, these regions represent a small portion of the Fermi surface when the SOC energies are much larger than the gap energies, so that an examination of the projected gap is still qualitatively useful in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Provided the superconducting state does not break time-reversal symmetry, the projected gap magnitude on band a can be found through [42] ˜∆2 ± = Tr[|{Hδ, ∆}|2P±] Tr[|Hδ|2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (5) where P±(k) = 1 2(1 ± Hδ(k)/εδ,k) which is a projection operator onto ± band by the energy dispersion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This projected gap magnitude is related to superconducting fitness [43, 44]: if it vanishes, the corresponding gap function is called unfit and will have a Tc = 0 in the weak coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Table III gives the projected gap functions for the pairing states discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The projection generally reduces the size of the gap, with the exception of the usual even-parity τ0ψk state (interestingly, the odd-parity τ0(dk ·σ) state has a gap that is generically reduced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This reduction strongly suppresses the Tc of the pairings state, where it enters exponentially in the weak-coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We later examine the different kp classes to identify fit gap functions since the Tc of these states will be the largest, given a fixed attractive interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' On the nodal plane, the projected gap functions, shown in Table III, simplify considerably since only ε0 and λk · ˆn are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For both type 1 and type 2 Hamiltonians, this leads to two gap functions that are fully fit, that is, not reduced by the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For type 1 Hamiltonians, these fully fit states are τ0ψk and τ3ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The state τ0ψk is even-parity and the state τ3ψk is odd-parity and, as discussed later, these two states play an important role in the appearance of a field-induced transition from even to odd parity superconductivity as observed in CeRh2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For gap functions described by vectors, for example dk, the projected gaps on the nodal plane are of the form |dk · ˆn|2 or |dk × ˆn|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is qualitatively different than the usual odd-parity single-band gap, where the gap magnitude is |dk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The latter requires that all three components of dk must vanish to have nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the projected gaps on the nodal planes, this requirement less stringent: only one or two components of dk need to vanish to have nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is closely related to the violation of Blount’s theorem on the nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Gap projection and the violation of Blount’s theorem Blount’s theorem states that time-reversal symmetric odd-parity superconductors cannot have line nodes when SOC is present [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Key to Blount’s theorem is the assumption that pseudsopsin shares the same symmetry properties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Type 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Type 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Gap function Inversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Gap projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Gap on nodal plane Inversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Gap projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Gap on nodal plane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ0ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ0(d · σ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='(t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2)|d|2 + |d · λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d · ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='(t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2)|d|2 + |d · λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d · ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ3ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|λ|2|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ3(d · σ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d · λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d · ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3|d|2 + |d × λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d × ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ1ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1|ψ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ1(d · σ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1|d|2 + |d × λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d × ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1|d|2 + |d × λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d × ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ2d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2|d|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|λ|2|d|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|d|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='τ2(ψ · σ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2|ψ|2 + |ψ × λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ × ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ · λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 + |λ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='|ψ · ˆn|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Classification of allowed pairing states for the kp theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For both type I and II TRIMs we give the symmetry under inversion, the gap projection onto the Fermi surface, and the gap on the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The momentum subscript indices k of the coefficient functions are omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' as usual spin [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' While the violation of Blount’s theorem in non-symmorphic space groups has been demonstrated earlier [18, 20, 21], here we present a simple proof that closely links anomalous pseudopsin to the violation of Blount’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The existence of anomalous pseudospin requires the presence of the translation mirror symmetry ˜ Mˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, the gap function can be classified as even or odd under this symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Momenta on the nodal plane are invariant under ˜ Mˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Hence, for these momenta, U † M∆(k)UM = ±∆(k) where the + (−) holds for a mirror-even (mirror-odd) gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For our basis choice UM = −iτβ(σ · ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Importantly, for both types the kp theories on the nodal plane are given by H(k) = ε0,k +iUM(λk · ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This defines the two bands E±(k) = ε0,k ±|λk · ˆn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Written in the band basis, we can divide the pairing potential into intraband and interband components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' On the nodal plane the intraband gap functions are explicitly given by P±∆P± = 1 4(−UM ± i sgn(λk · ˆn)){UM, ∆} , (6) while the interband components are P±∆P∓ = 1 4(−UM ± i sgn(λk · ˆn))[UM, ∆] (7) We observe that since a mirror-even gap function satisfies [UM, ∆] = 0, the interband gap components must vanish on the nodal plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' the pairing only involves particles from the same band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The general form of the BdG energy dispersion relation is then ±′ � (ε0,k ± |λk · ˆn|)2 + |∆±±|2 , (8) where intraband gap magnitude |∆±±|2 = 1 4Tr[|P±∆P±|2] and ±′ is the particle-hole symmetry index which is independent of band index ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since there is no requirement that |∆±±|2 = 0, line nodes are therefore not expected on the nodal plane, but rather we should generically find two-gap behavior with different size gaps on the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In contrast, for the mirror-odd gap functions we have {UM, ∆} = 0, so there is no intraband pairing on the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The eigenenergies for this interband pairing state are then ±′ � ±|λk · ˆn| + � ϵ2 0,k + |∆±∓|2 � , (9) where intraband gap magnitude |∆±∓|2 = 1 4Tr[|P±∆P∓|2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The gap has line nodes provided |λk · ˆn|2 > |∆±∓|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This result depends only on the mirror-odd symmetry of the gap, and not on the parity symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since gaps which are odd under both mirror and parity symmetry are allowed, this result shows that odd-parity gaps can have line nodes, thus demonstrating a violation of Blount’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 9 The origin of these nodes due to purely interband pairing implies that the nodes are shifted off the Fermi surface [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the spin-orbit coupling is too weak, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' |λk · ˆn|2 < |∆±∓|2, the nodes can annihilate with each other and are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This possibility has been discussed in the context of monolayer FeSe [46] and UPt3 [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The analysis above is valid even when Dirac lines pass through the TRIM points, as is the case in most of the derived kp theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' On the Dirac lines, the condition |λk · ˆn|2 < |∆±∓|2 must occur and the spectrum is therefore gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the Appendix A we present exact expressions for the energy eigenstates on the nodal plane for all possible combinations of mirror and parity gap symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Unconventional pairing states from electron-phonon interactions To highlight how pairing of anomalous pseudospin can differ from the single-band superconductivity, it is instructive to consider an attractive U Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Such a model is often used to capture the physics of electron-phonon driven s-wave superconductivity in single-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we show that this coupling also allows unconventional pairings states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, odd-parity states in type 1 kp Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Such a state has recently likley been observed in CeRh2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we consider a local Hubbard-U attraction on each site of the lattice and do not consider any longer range Coulomb interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These sites are defined by their Wyckoff positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Importantly, for the non-symmorphic groups we have considered here, each Wyckoff position has a multiplicity greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we limit our discussion to Wyckoff positions with multiplicity two, which implies that there are two inequivalent atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' An attractive U on these sites stabilizes a local spin-singlet Cooper pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since there are two sites per unit cell this implies that there are two stable superconducting degrees of freedom per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These two superconducting states can be constructed by setting the phase of Cooper pair wavefunction on each site to be the same or opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since only local interactions are included, both these two states will have the same pairing interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The in-phase state is a usual s-wave τ0ψk state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Identifying the other, out of phase, superconducting state requires an understanding of the relationship between the basis states for the kp Hamiltonians and orbitals located at the Wyckoff positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In general, this will depend on the specific orbitals included in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, the condition that the resultant pairing states must be spin-singlet and local in space (hence momentum independent) allows only two possibilities for this additional pairing state: it is either a τ1ψk or a τ3ψk pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Of these states, for two reasons, the τ3ψk state for type 1 Hamiltonains is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first reason is that this state is odd-parity and therefore offers a route towards topological superconductivity [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The second reason is that of the four possible states (τ1ψk or τ3ψk for type 1 or type 2 Hamiltonians), this is the only state that is fully fit on the nodal plane (as can be seen in Table III, the other three states have zero gap projection on the nodal plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This implies that for type 1 Hamiltonians, the odd-parity τ3ψk and the s-wave τ0ψk states can have comparable Tc since they both have the same pairing interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In practice, the τ3ψk state will have a lower Tc than the τ0ψk state since it will not be fully fit away from the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Table III reveals that this projection is given by the ratio |λk|2/(t2 1,k +t2 2,k +|λk|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For classes Dtype1 2h,1 , Dtype1 4h,1 , Dtype1 4h,3 , and Dtype1 4h,5 , this ratio is nearly one since the SOC terms are the largest in the kp Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This suggests that these classes offer a promising route towards stabilizing odd-parity superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We stress that because |λk|2/(t2 1,k + t2 2,k + |λk|2) is slightly less than one, the Tc of the odd-parity τ3ψk will be comparable but less than that of the usual s-wave state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, as we discuss later, the τ3ψk state can be stabilized over the usual s-wave τ0ψk state in an applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The identification of classes Dtype1 2h,1 , Dtype1 4h,1 , Dtype1 4h,3 , and Dtype1 4h,5 that maximize the Tc of odd-parity pairing from electron-phonon interactions allows the earlier theory for a field induced even to odd parity transition CeRh2As2 [9] (with space group 129) to be generalized to many other space groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' While the above odd-parity state is only relevant for type 1 Hamiltonians, for type 2 Hamiltonians, the usual s-wave interaction can develop a novel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, for the classes Ctype2 2h,2 and Dtype2 2h,4 , Table II shows that the state τ2σy is maximally fit and has s-wave symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, this state will admix with the usual s-wave τ0ψ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The theory describing this admixture formally resembles that of a Hund pairing mechanism proposed to explain the appearance of nodes in the likely s-wave superconductor KFe2As2 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The results of this analysis and a follow up analysis [51] allows some of the properties of this state to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' An important conclusion of these works is that an s-wave superconducting state can emerge even when pairing for the usual s-wave state is repulsive (that is for the Hubbard U > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This holds if two conditions are met: the effective interaction for the τ2σy state is attractive (to first approximation, this effective interaction does not depend upon U [50, 51]) and the two bands that emerge in the kp theory both cross the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This s-wave pairing state naturally lead to nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' ROLE OF MAGNETIC FIELDS The role of anomalous pseudopsin is perhaps most unusual in response to magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In many superconductors, there has been a push to drive up the magnetic field at which these are operational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Ising superconductors are one class of materials for which this has been successful, the in-plane critical field far surpasses the Pauli field, opening the door to applications [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Another relevant example is the field induced transition from an even parity to an odd-parity state observed in CeRh2As2 [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Recently, a powerful method to examine the response of superconductors to time-reversal symmetry-breaking fields has been developed by the projection onto the band-basis[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The form of the kp theories we have developed allows for the direct application of this projection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The response of superconductivity to time-reversal symmetry- breaking is described by a time-reversal symmetry-breaking interaction Hh(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A common form of TRSB Hamiltonian, and the one we emphasize here, is the Zeeman field interaction term, which is represented by Hh(k) = τ0(h · σ) , (10) where h is a magnetic field parameter in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that our qualitative results apply to a broader range of TRSB Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, this is true if the TRSB field shares the same symmetry properties as a Zeeman field (for example if Hh(k) describes the coupling between orbital angular momentum and an applied field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The theory introduces two parameters that quantify the response of superconductivity to time-reversal symmetry- breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first parameter is an effective g-factor given by ˜g2 ±,k,h = 2Tr[|{Hδ, Hh}|2P±] Tr[|Hδ|2]Tr[|Hh|2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (11) The second parameter is the field-fitness, given by ˜F±,k,h = Tr[|{{Hδ, ˜∆}, {Hδ, Hh}}|2P±] 2Tr[|{Hδ, Hh}|2P±]Tr[|{Hδ, ˜∆}|2P±] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (12) This field-fitness function ranges in value from zero to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When the field-fitness is zero, the superconducting state is not suppressed by the time-reversal symmetry breaking perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' With these two parameters, the response of superconductivity to applied fields and the temperature dependence of magnetic susceptibility in the superconducting state can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' With the choice of the time-reversal symmetry-breaking field as the Zeeman field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 10, one finds ˜g2 ±,k,h = t2 1,k + t2 α,k + (λk · ˆh)2 t2 1,k + t2 α,k + λ2 k (13) where α is a type index that is 2 for type 1 and 3 for type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This agrees with results in [53] derived for Hamiltonians that resemble type 1 Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that the band index ± and the magnitude of field h in the field-fitness and the g-factor do not change the outcome, thus they will be omitted in the subsequent sections and they will be denoted by ˜F 2 k,ˆh and ˜g2 k,ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Even parity superconductors It can be shown that the field-fitness parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 12 is 1 for all even parity states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, the magnetic response is governed solely by the generalized g-factor given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For momenta on the nodal plane, where t1,k = t2,k = t3,k = λk × ˆn = 0, the g-factor vanishes for magnetic fields orthogonal to ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is a direct consequence of the anomalous pseudospin, since the symmetries of the Pauli matrices formed from anomalous pseudospin do not allow any coupling to a Zeeman field perpendicular to ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' An immediate consequence is that superconductivity survives to much stronger fields than expected for these field orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, momenta that do not sit on the nodal plane also contribute to the superconducting state and their contribution needs to be included as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To quantify this, we solve for the Pauli limiting field within weak coupling theory at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For an isotropic s-wave superconductor, we find ln hP,ˆh h0 = −⟨ln |˜gk,ˆh|⟩k (14) 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 Γ X M (a) Energy (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 Γ X M (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' DFT bands of BiS2 near the X point (a) without and (b) with the SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The bands highlighted in the box are our focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' for field along direction ˆh, where h0 is the usual Pauli limiting field (found when the SOC is ignored), and ⟨·⟩k means an average over the Fermi surface weighted by the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Below, we apply this formula to BiS2-based superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that the spin susceptibility in the superconducting state can also be expressed using ˜gk,ˆh as well [42], and this shows that a non-zero spin susceptibility is predicted at zero temperature whenever the critical field surpasses h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Enhanced in plane field Pauli for BiS2-based superconductors Here we turn to recent experimental results on BiS2-based superconductors [22, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This material has the tetragonal space group 129 (P4/nmm) and it exhibits two electron pockets about the two equivalent X points [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When S is replaced with Se, it has been observed that the in-plane upper critical field surpasses the usual Pauli limiting field by a factor of 7 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' While it has been suggested that the local non-centrosymmetric structure is the source of this large critical field [54], there has been no quantitative calculation for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 14 to the kp theory at the X-point to see if it is possible to account for this large critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The X point in space group 129 belongs to class Dtype1 2h,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='For BiS2, the dispersion is known to be strongly two-dimensional (2D) [22, 55] so we consider the kp theory in the 2D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This kp theory is HBiS2 = ¯h2 2m � k2 x + γ2k2 y � − µ + t2kyτ2 + λxkyτ3σx + λykxτ3σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (15) Assuming s-wave superconductivity and accounting for the two equivalent pockets yields hP,ˆx = h0 � t2 2 + λ2x + |γλy| � |t2| + |γλy|(t2 2 + λ2x)1/4 (16) where h0 is the usual Pauli limiting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For simplicity we consider γ = 1 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 16 reveals that a large enhancement of the limiting field is possible and requires two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first is that t2 << λx, λy and second is that these is substantial anisotropy in λx and λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To understand if these conditions are reasonable, we have carried out density-functional theory (DFT) calculations on LaO1/2F1/2BiS2 with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' DFT calculations for LaO1/2F1/2BiS2 were carried out by the full-potential linearized augmented plane wave method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The Perdew- Burke-Ernzerhof form of the exchange correlation functional [57], wave function and potential energy cutoffs of 14 and 200 Ry, respectively, muffin-tin sphere radii of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='0 ˚A for Bi, S, La, O atoms, respectively, the experimental lattice parameters [58], and an 15 × 15 × 5 k-point mesh were employed for the self-consistent field calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The virtual crystal approximation was used by setting the nuclear charge Z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='5 at O(F) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The resultant bands are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Without SOC, the band splitting along Γ to X yields an estimate for t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When SOC is present, the band splitting along the X to M yields λy and the band splitting along Γ to X yields � λ2x + t2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The DFT calculated splittings suggest that λx is the largest parameter by a factor of 3-4, while t2 and λy are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This suggests that the conditions to achieve a large critical field are realistic in BiS2-based superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Note that the largest 12 observed Pauli fields are found when the S is substituted by Se [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Se has a larger SOC than S, suggesting that the λi parameters will be increased from what we estimate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is currently under exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is worthwhile contrasting the above theory with that for Fe-based materials in which electron pockets exist near the M point of space group 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The M-point is described by class Dtype1 4h,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this case, an analysis similar to to BiS2 gives an enhancement of only √ 2 of the Pauli field for in-plane fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For c-axis fields, this class implies a significantly enhanced Pauli limiting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These results are consistent with experimental fits to upper critical fields in Fe-based superconductors that reveal that the upper critical field for in-plane fields are Pauli suppressed while those for field along the c-axis are not [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The contrast bewteen Fe-based materials and BiS2-based materials highlights the importance of the different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In particular, the lower orthorhombic symmetry of the X point allows protection to in-plane fields not afforded to the M point, where the theory is strongly constrained by tetragonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Pair density wave states In BCS theory, a spin-singlet superconductor is suppressed by the Zeeman effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Under a sufficiently strong magnetic field, the pairing susceptibility can be peaked at non-zero Cooper pair momenta, leading to a pair density wave or FFLO state [60–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A schematic phase diagram for a centrosymmetric system is shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The typically first order phase transition (double solid line) between the uniform and FFLO state ends at a bicritical point (Tb, Hb), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' FFLO state only exists for T < Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A weak-coupling calculation reveals that for the usual FFLO phase, Tb/Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='56 It is known that for locally non-centrosymmetric superconductors, FFLO-like phases can appear at lower fields Hb and higher temperatures Tb than the usual FFLO-like instability [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is closely linked to the symmetry required instability to a pair density wave state for non-centrosymmetric superconductors when a field is applied [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For a non-centrosymmetric system under magnetic field, both inversion and time-reversal symmetry are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As a result, the pairing susceptibility is generically peaked at non-zero momentum and Tb = Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For locally non-centrosymmtric superconductors, inversion symmetry is locally broken on each sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In an extreme case, if the two sublattices are decoupled, then the system effectively becomes non-centrosymmetric, and under a small magnetic field, an FFLO state can exists right below the zero-field superconducting Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, these sublattices are generically coupled so that Tb = Tc is not realized in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we show that for type 1 Hamiltonians, FFLO-like states can in principle exist up to Tb = Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Schematic phase diagram for a spin-singlet superconductor under Zeeman effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Single solid lines denote continuous phase transitions while double solid lines denote first-order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To show this, we consider the 2D version of class Dtype1 4h,1 and use the pairing susceptibility to calculate Tb and Hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In 2D, class Dtype1 4h,1 has the following normal state Hamiltonian: HD4h,1 = ¯h2 2m(k2 x + k2 y) − µ + t1kxkyτ1 + λxτ3(kyσx + kxσy) + Hxσx (17) λx denotes the strength of the local inversion symmetry breaking (local Rashba SOC), while t1 is the inter-sublattice 13 coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The pairing susceptibility for an s-wave state with gap function τ0ψk is χpairing(Q) = − 1 β � ωn � (p,p+Q)∈FS Tr [G0(Q + p, ωn)G0(p, ωn)] , (18) where G0 is the normal state Green’s function written in Nambu space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The FFLO state is favored, if the pairing susceptibility is peaked at non-zero Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We examine the position of the bicritical point (Tb, Hb), as a function of λx/(t1kF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We use the following two equations to locate the bicritical point: (1) The bicritical point lies on the BCS transition for the uniform superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (2) The bicritical point is a continuous phase transition between uniform and FFLO superconductivity, where ∇2 Qχpairing(Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The result is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1000 × 1000 points are sampled in the 2D Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Other parameters are t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2, t = µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' An energy cutoff of Ec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 is applied to determine the position of the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 x/t 1/kF 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 1 Tb/Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 x/t 1/kF 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 1 Hb/Hb( x=0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The position of the bicritical point (Tb, Hb), as a function of λx/kF t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These results show that for zero λx/kF t1, a usual FFLO phase is found (that is Tb/Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As the SOC λx increases or equivalently, as kF decreases, Tb increases and approaches the zero-field critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the meantime, Hb monotonically decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We have shown that the FFLO phase can exist up to Tb = Tc for a 2D version of class Dtype1 4h,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Key is that SOC is the leading order term in the kp theory and this is also the case for other type 1 Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Hence the optimal conditions for an enhanced FFLO phase to occur are when fields are applied in-plane (perpendicular to the c-axis) for classes Dtype1 2h,1 , Dtype1 4h,1 , Dtype1 4h,3 , and Dtype1 4h,5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Odd-parity superconductors For odd parity superconductors, the field fitness parameter ˜Fk,ˆh can become less than 1 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Of particular interest is when ˜Fk,ˆh = 0 since this implies that Tc is unchanged by the time-reversal symmetry breaking field (this is independent of the effective g-factor) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For anomalous pseudospin this possibility leads to two consequences not expected for spin-triplet states made from usual spin-1/2 fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The first is a field induced transition from an even to an odd parity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The second is that, in spite of the presence of strong SOC, the superconducting state is immune to magnetic fields for all field orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We discuss these each in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Field induced even to odd parity transitions In CeRh2As2, a field induced even to odd parity transition has been observed for the field oriented along the c-axis in this tetragonal material [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Earlier, we argued that this was due the anomalous pseudospin that arises on the Brillouin zone faces in the non-symmorphic space group P4/nmm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we show how this can be generalized to other space groups that admit type 1 kp theories and determine which classes are optimal for observing such a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed in Section IV C, an attractive electron-phonon like interaction gives rise to both both a usual 14 s-wave τ0ψk state and an odd-parity τ3ψk state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These two states have the same pairing interaction, but the gap projected onto the band basis is generally smaller for the τ3ψk state than for the τ0ψk state, implying that τ0ψk state has the higher Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the type 1 classes Dtype1 2h,1 , Dtype1 4h,1 , Dtype1 4h,3 , and Dtype1 4h,5 , anomalous pseudospin leads to Tc’s that are nearly the same for the even τ0ψ and odd-parity τ3ψ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' These classes are therefore promising for observing a field induced transition from an even-parity to an odd-parity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To determine if a such a field induced transition occurs we compute ˜Fk,ˆh for a pairing state ˜∆ = τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We find for type 1 kp theories ˜Fk,ˆh = (ˆh · λk)2(t2 1,k + t2 2,k + |λk|2) |λk|2[ˆh2(t2 1,k + t2 2,k) + (ˆh · λk)2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (19) Notice if ˆh · λk = 0, then ˜Fk,ˆh = 0 which maximizes Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To determine the field orientations for which ˜Fk,ˆh = 0, we examine the form of λk in the type 1 classes discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In all these classes, the λz,k component appears with a higher power of momenta than the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, the field should be applied along the ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As an example, consider the class Dtype1 4h,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here λz,k ∝ kxkykz(k2 x − k2 y) while λx,k ∝ ky and λy,k ∝ ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this case λk will be in-plane to an excellent approximation, and an even to odd-parity transition can be expected for the field along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, classes Dtype1 2h,1 , Dtype1 4h,1 , Dtype1 4h,3 , and Dtype1 4h,5 and, hence, space groups 56, 58, 59, 62, 128, 129, 130, 136, 137, and 138 are promising for realizing a field-induced even to odd parity transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Field immune odd-parity superconductivity For a conventional spin-triplet superconductor (with ∆ = dk · σ) formed from usual spin-1/2 pseudospin, SOC typically pins the direction of the vector dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the applied field is perpendicular to dk, that is if dk · ˆh = 0, then the Tc for this field orientation is unchanged [63–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since there exists at least one field direction for which dk · ˆh ̸= 0, it is not expected that usual spin-triplet superconductors are immune to fields applied in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For anomalous pseudopsin, this is not the case, it is possible for an odd-parity state to be robust against suppression for arbitrarily oriented magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To show how this is possible, we calculate ˜Fk,ˆh for ∆ = τ0(dk · σ) for type 1 kp theories, this yields ˜Fk,ˆh = [(t2 1,k + t2 2,k)dk · ˆh + (dk · λk)(λk · ˆh)]2 [(t2 1,k + t2 2,k)ˆh2 + (λk · ˆh)2][(t2 1,k + t2 2,k)|dk|2 + (dk · λk)2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (20) We first note that near the nodal plane, the effective g-factor is small for in-plane fields ˆn·⃗h = 0, so that for these field orientations superconductivity is not strongly suppressed (this is true for both even and odd-parity superconducting states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Hence, to show that an odd-parity state survives for all field orientations, we need to show that ˜Fk,ˆh ≈ 0 for a field applied along the nodal plane normal where λk · ˆh becomes maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Near the plane we expect that λk · ˆh ≫ � t2 1,k + t2 2,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Also, (t2 1,k + t2 2,k) is small compared to λ2 k, so ˜Fk,ˆh is dominated by the dk · λk term in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Hence if the denominator |t1,2dk| is much bigger than dk ·λk, then ˜Fk,ˆh ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Given that λˆn is the largest SOC component, this requirement is equivalent to λ⊥ ≪ t1,2 and dk ⊥ ˆn (where λ⊥ is the magnitude of the SOC perpendicular to ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As a relevant example of the above mechanism we consider UPt3 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The superconducting state in UPt3 is believed to be an E2u state, with order parameter ∆ = ηp(σxky + σykx) + ηfσzkzkxky (we only include one component of this two-component order parameter since similar arguments hold for the second component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In general, since the p-wave and f-wave components have the same symmetry, both ηp and ηf are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, theories based on the usual pseudospin typically require ηp = 0 due to the experimental observations discussed below [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Below we further show that ηp = 0 is not required for these experimental observations when anomalous pseudospin is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Indeed, these experiments are consistent with ηf = 0 and ηp ̸= 0 if pairing occurs predominantly near the nodal plane kz = π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Thermal conductivity experiments suggest the existence of line nodes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For usual pseudospin, the state σxky + σykx is either fully gapped or has only point nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is one reason to expect that ηp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, as illustrated in Table II, line nodes are expected for this state on the kz = π/c plane (note this conclusion also follows from Refs [18, 19, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is relevant for UPt3 since it is known to have the ‘starfish’ Fermi surface near this nodal plane [24] which belongs to class Dtype1 6h In terms of paramagnetic suppression, the superconducting state is known to be more robust under B ⊥ z compared to B ∥ z [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the usual pseudospin, this requires dk ∥ z, and thus ηp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, on the ‘starfish’ Fermi 15 surface, the small g-factor for B ⊥ z can serve to protect the p-wave state against paramagnetic suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As discussed above, the suppression from B ∥ z depends on the ratio λx,y/t1,2, while the g-factor for B ⊥ z depends on the ratio (t1,2, λx,y)/λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The requirement λx,y/t1,2 > (t1,2, λx,y)/λz is thus sufficient to match the observations on the upper critical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If both ratios are much smaller than one, the p-wave state is immune to paramagnetic suppression for field along arbitrary directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This could be relevant to the approximately unchanged Knight shift in the superconducting state [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that the use of ˜Fk,ˆh to determine the magnetic response relies on the validity of projection to a single band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, for class Dtype1 6h band degeneracies exist along three Dirac lines for which this projection is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In Appendix B we include a detailed numerical calculation that includes interband effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 8-FOLD DEGENERATE POINTS: APPLICATION TO UCOGE The arguments presented above relied on the 4-fold degeneracy at TRIM points when SOC is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, some of these TRIM points have an 8-fold degeneracy without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' It is reasonable to ask if the conclusions found for kp theories of 4-fold degenerate points discussed above survive to 8-fold degenerate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To address this, we have determined the symmetries of all orbital operators in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We find that in most cases, the 8-fold degeneracy at these TRIM is split by a single SOC term of the form Oσi where O is a momentum independent 4 by 4 orbital matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='IV, we give the direction of the spin component σi that appears in this SOC term at the TRIM point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The existence of this single SOC term ensures small effective g-factors for fields perpendicular to the spin-component direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Consequently, the conclusions associated with the effective g-factor anisotropy discussed in Section V still hold for these 8-fold degenerate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We note that the 8-fold degeneracy at the A point of space groups 130 and 135 are not split by SOC and these points provide examples of double Dirac points examined in [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Spin Alignment Space Group Momenta σx 54(U1U2),54(R1R2),56(U1U2),60(R1R2),61(S1S2),62(S1S2),205(M1M2) σy 52(S1S2),56(T1T2),57(T1T2),57(R1R2),61(T1T2),130(R1R2),138(R1R2) σz 60(T1T2),60(U1U2),61(U1U2),62(R1R2),128(A3A4),137(A3A4),176(A2A3),193(A3),194(A3) TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Spin alignment of 8-fold degenerate TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' One material for which these 8-fold degenerate points are likely to be relevant is the ferromagetic superconductor UCoGe, which crystalizes in space group 62 (Pnma) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' UCoGe is believed to be a possibly topological odd-parity superconductor [17, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Our Fermi surface (given in Figure 3) reveals that all Fermi surface sheets lie near nodal planes with anomalous pseudospin and further reveal tube-shaped pockets that enclose the zone-boundary S point and stretch along the S-R axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we focus on these Fermi surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This feature reasonably agrees with previous works [72–74] using local density approximation and the existence of these tube shaped Fermi surfaces is consistent with quantum oscillation measurements [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here density-functional theory calculations for UCoGe were carried out by the full-potential linearized augmented plane wave method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Perdew-Burke-Ernzerhof form of exchange correlation functional [57], wave function and potential energy cutoffs of 16 and 200 Ry, respectively, muffin-tin sphere radii of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 ˚A for U and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 ˚A for Co and Ge, respectively, the experimental lattice parameters [76], and an 8 × 12 × 8 k-point mesh were employed for the self-consistent field calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Spin-orbit was fully taken into account in the assumed nonmagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Fermi surface was determined on a dense 30 × 50 × 30 k-point mesh and visualized by using FermiSurfer [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Both the R and S points are 8-fold degenerate TRIM when SOC is not included for space group 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Interestingly, from Table IV, the effective g-factors for fields along ˆy and ˆz directions are zero at the S-point and are zero for fields along ˆx and ˆy directions at the R-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This indicates that superconductivity (both even and odd-parity) on the tube-shaped Fermi surfaces will be robust against magnetic fields applied along the ˆy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is the field direction for which the upper critical field is observed to be the highest and for which an unusual S-shaped critical field curve appears [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We leave a detailed examination of the consequences of anomalous pseudospin in space group 62 on superconductivity to a later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' CONCLUSIONS Non-symmorphic symmetries allow the existence of nodal planes at Brillouin zone edges when no SOC is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' When SOC is added, the pseudospin on these nodal planes has different symmetry properties than usual pseudospin- 16 X U Z Y S R !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' DFT Fermi surface of UCoGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here we have classified all space groups and effective single-particle theories near TRIM points on these nodal planes and examined the consequences of this anomalous pseudospin on the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We have shown how this enhances the Tc for odd-parity superconducting states due to attractive interactions, leads to unexpected superconducting nodal properties, allows large Pauli limiting fields and pair density wave states for spin-singlet superconductors, gives rise to field immune odd-parity superconductivity, and to field driven even to odd-parity superconducting transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' While we have emphasized nodal planes on which anomalous pseudospin exists, there are also materials for which anomalous pseudospin develops on nodal lines and not on nodal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Some such materials also exhibit unusual response to magnetic fields [78–80], suggesting a broader range of applicability for anomalous pseudospin superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' ACKNOWLEDGEMENTS DFA, HGS, and YY were supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0021971 and by a UWM Discovery and Innovation Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' MW and TS were supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0017632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' PMRB was supported by the Marsden Fund Council from Government funding, managed by Royal Society Te Aparangi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We acknowledge useful discussions with Mark Fischer, Elena Hassinger, Seunghyun Khim, Igor Mazin, and Manfred Sigrist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Manchon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Koo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Nitta, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Gu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Ma, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Ren, Superconductivity with a Violation of Pauli Limit and Evidences for Multigap in η-Carbide Type Ti4Ir2O, Chinese Physics Letters 39, 027401 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 20 Appendix A: Full excitation spectrum on the nodal plane On the nodal plane, the Bogoliubov de-Gennes Hamiltonian takes the form H = � k Ψ† k � ε0,k + τ3(λk · ˆn)(σ · ˆn) ∆k ∆† k −ε0,k − τ3(λk · ˆn)(σ · ˆn) � Ψk, (A1) It is possible to classify the gap symmetry as even or odd under both inversion and mirror symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For momenta on the nodal surface we have, U † P ∆kUP = ± ∆−k U † M∆kUM = ± ∆k (A2) where for type 1 TRIM UP = τ1 and UM = −iτ3σz and for type 2 TRIM UP = τ0 and UM = −iτ2σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We label the gaps as ∆1(2),i,j where i = ± labels the parity symmetry and j = ± labels the mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Here, for clarity, we drop the k labels (note that k is unchanged by the mirror symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the type 1 TRIM, we write the gap functions in terms of the complete set of gap functions with the correct symmetries given in Table III as ∆1,++ = ψ0τ0 + (dz · ˆn)(σ · ˆn)τ3 ∆1,+− = ψxτ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2 ∆1,−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2 ∆1,−− = (d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (ψ · ˆn)(σ · ˆn)τ2 (A3) where di are odd functions of k and ψi are even functions of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A1, the corresponding quasiparticle excitation energies can be found to be E1,++ = ±′ � (ϵ0 ± λ · ˆn)2 + (ψ0 ± dz · ˆn)2 E1,+− = ±′ �� ϵ2 0 + ψ2x + (dz × ˆn)2 + d2y ± λ · ˆn � E1,−+ = ±′ � (ϵ0 ± λ · ˆn)2 + (ψz ± d0 · ˆn)2 + (dx × ˆn)2 + (ψ × ˆn)2 ± 2(dx × ψ) · ˆn E1,−− = ±′ �� ϵ2 0 + (d0 × ˆn)2 + (dx · ˆn)2 + (ψ · ˆn)2 ± λ · ˆn � (A4) where the prime denotes independent choices of the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For type 2 TRIM we similarly have ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='++ = ψ0τ0 + (ψ · ˆn)(σ · ˆn)τ2 ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+− = ψxτ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2 ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2 ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='−− = (d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (dz · ˆn)(σ · ˆn)τ3 (A5) The quasiparticle excitation spectra for these states are E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='++ = ±′ � (ϵ0 ± λ · ˆn)2 + (ψ0 ± ψ · ˆn)2 E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='+− = ±′ �� ϵ2 0 + ψ2x + ψ2z + (ψ × ˆn)2 ± λ · ˆn � E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='−+ = ±′ � (ϵ0 ± λ · ˆn)2 + (dy ± d0 · ˆn)2 + (dx × ˆn)2 + (dz × ˆn)2 ± 2(dx × dz) · ˆn E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='−− = ±′ �� ϵ2 0 + (d0 × ˆn)2 + (dx · ˆn)2 + (dz · ˆn)2 ± λ · ˆn � (A6) 21 Appendix B: Magnetic susceptibility UPt3 In the main text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' we illustrated how the p-wave state in UPt3 is immune to the magnetic field along arbitrary directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' An important step is to consider the small g-factor for field B ⊥ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, the discussion is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the normal state, there exist 4-fold degenerate Dirac lines on the plane kz = π/c, where the g-factor is not small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In terms of the field fitness, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='20 in the main text only considered doubly degenerate bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In principle, extra terms in the field fitness are needed for to describe these Dirac lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' However, the Fermi surface is not right on the nodal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This can make the Dirac lines unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this section, we will explicitly check the field response in the superconducting state through a numerical calculation on a tight-binding model for UPt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In the following calculations, we will focus on the Knight shift (spin-susceptibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Knight shift measures spin polarization at atom sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' By extracting spin susceptibility χs, one can determine pairing functions of an uncon- ventional superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For a single-band spin-triplet superconductor, the change of Knight shift depends on the orientation of magnetic field with respect to the d-vector of the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the magnetic field is per- pendicular to the d-vector, the Knight shift should be a constant across superconducting Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the magnetic field is parallel to the d-vector, the Knight shift will decrease to zero as temperature approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For the multi-band non-symmorphic superconductor UPt3, Knight shift is almost unchanged for all field orientations, suggesting the importance of spin-orbit coupling in this heavy fermion material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' One of the Fermi surfaces (‘starfish’) of UPt3 is flat and located near the high symmetry plane kz = π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Zeeman terms Bxσx and Byσy then becomes inter-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' From non-generate perturbation theory, spin susceptibilities are inversely proportional to the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This is different from the intra-band Zeeman effect, where susceptibilities are proportional to the density of states on Fermi surface, according to degenerate perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Since the superconducting gap is much smaller than the band gap, inter-band susceptibilities will be unchanged across Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the superconductivity is mainly developed on the above flat Fermi surface, then Knight shift is expected to be unchanged for in-plane magnetic fields, regardless of the superconducting pairing symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' If the d-vector is in-plane, then Knight shift will also be unchanged for a perpendicular magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this section, we will explicitly illustrate this idea to understand the experimental results on UPt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Crystal structure of UPt3 with the unit vector e1 = (1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The 4 × 4 normal state Hamiltonian reads [67]: H = ε(k) + gz(k)σzτ3 + a1(k)τ1 + a2(k)τ2 + [gx(k)σx + gy(k)σy] τ3 εk = 2t � i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 cos k∥ · ei + 2t3 cos kz − µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' gz(k) = gz0 � i sin k∥ · ei a1(k) = 2t′ sin kz 2 � i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 sin k∥ · ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' a2(k) = 2t′ sin kz 2 � i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='3 cos k∥ · ri gx(k) = gx0fxfy sin kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' gy(k) = gy0(f 2 x − f 2 y ) sin kz fx ≡ sin k∥ · e1 − sin k∥ · e2 + sin k∥ · e3 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' fy ≡ √ 3 sin k∥ · e2 − sin k∥ · e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (B1) here (kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' kz) are relative to the high symmetry point (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Relevant vectors ei and ri can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' τi matrices live in the sublattice space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' On the high-symmetry plane kz = 0, the inter-sublattice hopping a1,2 and the spin-flip SOC gx,y vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' |k, m = 1, ↑⟩ and |k, m = 2, ↓⟩ states form a pseudospin band, while |k, m = 2, ↑⟩ and |k, m = 1, ↓⟩ states form another band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 22 We now study spin susceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We will focus on a p-wave state in the E2u channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Its d-vector is in-plane: d = ∆(T)(fx, −fy, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' fx and fy are introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='B1, and they transform as kx and ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The gap magnitude is taken to be ∆(T) = ∆0 � 1 − T/Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' t = 1, t3 = −4, gz0 = 2, µ = 12 and ∆0 = Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='001 is taken in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 1 T/Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content="1 t'=0, gx0=gy0=0 x z 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 1 T/Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content="1 t'=0, gx0=gy0=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1 x z x,inter z,inter 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='8 1 T/Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content="1 t'=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='1, gx0=gy0=0 x z x,inter z,inter FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Spin susceptibilities as a function of temperature, for (left) a1 = a2 = gx = gy = 0, which would be the case if the Fermi surface exactly lied on the high-symmetry plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (middle) non-zero spin-flip SOC but zero inter-sublattice hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' (right) non-zero inter-sublattice hopping but zero spin-flip SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' To illustrate the effect of the anomalous pseudospin, we start with a toy model with zero inter-sublattice hopping and spin-flip SOC: t′ = gx0 = gy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The corresponding four terms vanish in the normal state Hamiltonian: a1 = a2 = gx = gy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In this extreme case, the spin susceptibilities are unchanged across Tc, as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We now turn on the spin-flip SOC (gx0 and gy0), while keeping the inter-sublattice hopping t′ to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' hxσx develops an intra-band component, which will be suppressed in the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' As a result, the total χx deep in the superconducting state starts to decrease as function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For χz, spin-flip SOC induces higher-order terms in the E2u channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The d-vector develops non-zero z-component in the band basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This causes a decrease in χz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The result for gx0 = gy0 can be found in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The inter-band susceptibilities in the normal state are included in dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' We now turn on the inter-sublattice hopping t′, while keeping the spin-flip SOC (gx0 and gy0) to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' A similar effect is expected for χx due to the intra-band contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' For χz, since σz is a good quantum number, χz will be unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The result can be found in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Experimentally, the superconducting state is known to be more robust under B ∥ x compared to B ∥ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' In other words, the decrease in χx needs to be smaller than χz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' This scenario is closer to the second limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' 23 Appendix C: 8-fold Representations Here, we list the symmetries of all orbital operators near the 8-fold degenerate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The point group that keeps the TRIM point invariant can be found in the title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' The bracket notation [·] is also used for antisymmetric operators which was τ2 in the main context, but in 8-fold cases, the antisymmtric component is not unique due to the higher degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Space group momenta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Point group D2h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='54(U1U2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='54(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='56(U1U2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='60(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + Au + 2B2u + B3u + [Ag] + [B3g] + [Au] + 2[B1u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='61(S1S2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='62(S1S2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='205(M1M2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='52(S1S2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='56(T1T2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='57(T1T2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='57(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='61(T1T2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='130(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='138(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='60(T1T2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + B1g + 2B2g + 2B3g + 2Au + B2u + B3u + [Ag] + [B1g] + 2[B1u] + [B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='60(U1U2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='61(U1U2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='62(R1R2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + B1g + 2B2g + 2B3g + 2B1u + B2u + B3u + [Ag] + [B1g] + 2[Au] + [B2u] + [B3u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Space group momenta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Point group D4h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='128(A3A4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='137(A3A4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Space group momenta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Point group C6h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='176(A2A3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Ag + Bg + E1g + E2g + Au + Bu + E1u + [Ag] + [Bg] + [Au] + [Bu] + [E2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Space group momenta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='Point group D6h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='193(A3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='A1g + B2g + E1g + E2g + A1u + B1u + E1u + [A2g] + [B1g] + [A2u] + [B2u] + [E2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='194(A3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='A1g + B1g + E1g + E2g + A1u + B2u + E1u + [A2g] + [B2g] + [A2u] + [B1u] + [E2u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content='TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} +page_content=' Symmetries of orbital operators at the 8-fold degenerate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQfJCwC/content/2301.11458v1.pdf'} diff --git a/Q9E4T4oBgHgl3EQfKgxv/content/2301.04930v1.pdf b/Q9E4T4oBgHgl3EQfKgxv/content/2301.04930v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..710bede060ef504224c1a31cfafbb1330095bf7e --- /dev/null +++ b/Q9E4T4oBgHgl3EQfKgxv/content/2301.04930v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3978d17e7be421a034d30ec488f6803d01064ff5096239ee5b9868ecb18809bc +size 1458774 diff --git a/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/2301.05365v1.pdf.txt b/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/2301.05365v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a7fc4852c5c5404fd11370033d4264e43f0ea0d --- /dev/null +++ b/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/2301.05365v1.pdf.txt @@ -0,0 +1,3987 @@ +Randomization Tests for Adaptively Collected Data +Yash Nair and Lucas Janson +Department of Statistics, Harvard University +Abstract +Randomization tests (including permutation tests) are one of the most fundamental methods in +statistics, enabling a range of inferential tasks such as testing for (conditional) independence of random +variables, constructing confidence intervals in semiparametric location models, and constructing (by +inverting a permutation test) model-free prediction intervals via conformal inference. Randomization +tests are intuitive, easy to implement, and exactly valid for any sample size, but their use is generally +confined to independent and/or exchangeable data. Yet in many applications including clinical trials, +online education, online advertising, and protein design, data is routinely collected adaptively, meaning +that the aspects of the data under the data collector’s control (e.g., treatment assignments) are assigned +at each time step via a (possibly randomized) algorithm that depends on all the data observed so far; +such assignment algorithms include (contextual) bandit and reinforcement learning algorithms as well as +adaptive experimental designs. In this paper we present a general framework for randomization testing on +adaptively collected data (despite its non-exchangeability), encompassing (and in some cases improving) +the few existing results on randomization testing and conformal inference for adaptively collected data, +as well as many other important settings. The key to our framework is the ability to compute likelihood- +ratio-based weights involving known quantities based purely on the known adaptive assignment algorithm, +as long as a certain proportionality condition is met. These weights can then be accounted for in our +framework to conduct an exact randomization test, but in order for the test to be powerful, resamples +need to be diverse yet have weights as close to equal as possible. Thus, we additionally present novel +computationally tractable resampling algorithms for various popular adaptive assignment algorithms, +data-generating environments, and types of inferential tasks. Finally, we demonstrate via a range of +simulations our framework’s power (in the case of hypothesis testing) and narrow widths (in the case of +confidence or prediction intervals produced by inverting randomization tests). +1 +Introduction +1.1 +Motivation +Randomization tests form an important methodological framework that enjoys a wide array of uses across +statistics, ranging from testing equality of distributions to testing (conditional) independence between co- +variates and response in supervised learning settings. Beyond these classical uses of randomization tests, +the framework also encompasses permutation testing and (inversely) conformal inference (Vovk et al., 2005). +We briefly note here that, throughout this paper, we use the phrase “randomization test” not to refer to a +test applied on data obtained via the physical act of randomization taken by an experimenter, but rather +the randomization used by the test itself to compute a p-value. Specifically, we view a randomization test +as a randomized procedure, taking the observed data set D as input, that samples m copies ˜D(1), . . . , ˜D(m) +that are jointly exchangeable with D under the null, and then computes a (valid) p-value by taking a simple +average of indicators comparing the resampled data to the observed data via a given test statistic. This +interpretation of a randomization test has been referred to as a quasi-randomization test by Zhang and Zhao +(2022). +While randomization tests can make very weak assumptions on the marginal distribution of observations, +they generally make quite restrictive assumptions on the joint distribution of the data. +1 +arXiv:2301.05365v1 [stat.ME] 13 Jan 2023 + +In particular, they usually require independent and/or exchangeable data (e.g., Fisher et al., 1937; Pitman, +1937; Lehmann et al., 2005; Vovk et al., 2005; Edgington and Onghena, 2007; Candes et al., 2018), an +assumption that is often violated in many real-world settings in which data are gathered in an adaptive +fashion. One such example is drug discovery research, in which a scientist adaptively submits experiments +serially, where the tth experiment’s design is based upon the results of the first t − 1 (Popova et al., 2018). +This complex dependency will, in general, violate the exchangeability assumption when analyzing the data +from all experiments at once. +Similar situations arise in experimental designs in the natural sciences more broadly in addition to mobile +health, adaptive clinical trials, online education, and online advertising, to name a few. +Scenarios like those above naturally lead to a number of important inferential tasks that need to be performed +on adaptively collected data, all of which we show can be performed via a randomization testing-based +framework: +• Testing if two or more treatments induce the same distribution over outcomes +• Detecting non-stationarity in the data (i.e., testing whether the outcome depends not only on the +treatment, but also on time) +• Using past data to predict the outcomes of future samples with quantifiable uncertainty +• Constructing confidence intervals for the difference in locations of outcome distributions corresponding +to different treatments in semiparametric models +We now briefly describe various settings in which the above tasks and generalizations thereof are both difficult +and important problems and motivate them with real-world examples. +Problem Domain 1 (Comparing response distributions of different covariates). An interesting and chal- +lenging problem in adaptive data collection settings is to test whether two or more different treatments +amongst a total of K give rise to the same response distribution, i.e., for some pair i, j ∈ {1, . . . , K}, is +(Y | X = i) +d= (Y | X = j)? In the most extreme case, one might ask if there is even any dependency +at all between X and Y (i.e., whether the treatment has any effect on the outcome). Beyond the mobile +health examples delineated below in Problem Domain 2, for which it is important to test if two different +treatments actually have the same effect (or whether treatments have any effect at all), such testing is also +useful in recommender systems (Schafer et al., 1999; Li et al., 2010), to determine if different content dis- +plays actually produce different outcomes. Further complicating this setting is that different users may react +differently (e.g., as measured by clickthrough rate) to different content. Thus, one might ask: Given the +‘context’ surrounding the user’s preferences, do they react the same way to different advertisements? +Problem Domain 2 (Testing non-stationarity). An important problem that arises in domains in which +actions are adaptively chosen and outcomes observed, is that of non-stationarity detection. That is, one may +ask if the conditional distributions Yt | Xt in data gathered via some adaptive decision-making procedure +change with t. One such scenario in which non-stationarity testing is important is in mobile health applica- +tions. For example, the Just-in-Time Adaptive Intervention (JITAI) (Nahum-Shani et al., 2016) is a mobile +health framework designed to provide appropriate support to patients with dynamic, time-varying states (e.g., +mood, location, etc.). JITAIs have been applied to, for instance, suicide prevention (Coppersmith et al., +2021), smoking relapse prevention (Battalio et al., 2021), and addiction science (Carpenter et al., 2020). In +situations like these, it is vital to be able to know whether or not a patient’s response, Yt, to the treatment, +Xt, is varying over time, t, so as to facilitate administration of the most effective and appropriate care +possible. Tests of non-stationarity provide a principled method of achieving this goal. +Problem Domain 3 (Predicting with high confidence). In experimental design settings, providing prediction +intervals for the results of future experiments can be instrumental in guiding researchers in decision-making. +A motivating example comes from Alvarsson et al. (2021). In their work, the authors introduce conformal +inference (Vovk et al., 2005; Shafer and Vovk, 2008; Vovk et al., 2009; Vovk, 2013; Burnaev and Vovk, 2014) +to researchers in the field of drug discovery and go on to give an example of how the methodology—which +2 + +also assumes i.i.d. or exchangeable data—can be applied to classify various types of ATP-Binding Cassette +transporters at any user-specified uncertainty (i.e., significance) level. However, in light of the sequential +and adaptive nature of many experimental designs in this field and in related fields like drug development +(e.g., Mahajan and Gupta, 2010; Godfrey et al., 2013; Pallmann et al., 2018), there is need for provably +valid test-time prediction intervals for these adaptively collected data. +Problem Domain 4 (Relating parameters in semiparametric models). A final difficult problem in scenarios +like those of the previous Problem Domains is to estimate how the parameters of different response distribu- +tions belonging to a semiparametric model relate to one another. For example, suppose that Y | X = x ∼ pθx, +where {pθ : θ ∈ Rd} is a location family with parameter indexed by the treatment. How can we estimate +θx − θx′ for treatments x and x′? This problem setting is ubiquitous, for example, in the multi-armed bandit +literature, where it is common to assume that the reward distributions of all arms are distributed accord- +ing to a location family like the Normal (Lai and Robbins, 1985). In such settings, being able to estimate +parametric relationships between different treatments in finite samples is useful for a variety of real-world +problems. As an example, as described in Zhang et al. (2020), performing post hoc inference can be useful +in many settings ranging from recommender systems (Mary et al., 2015) to anomaly detection (Ban and He, +2020) to finance and portfolio selection (Huo and Fu, 2017). In such settings, after-the-fact inference may be +helpful to researchers and practitioners who wish to understand the differences between different treatments +and potentially utilize such differences to guide future decision-making. +A number of tasks that we consider in this paper—and, indeed, some of those presented in the Problem +Domains above—involve the construction of prediction or confidence intervals. However, as we show, con- +structing such intervals reduces to hypothesis testing by inverting the corresponding test. In particular, +although not usually framed this way, conformal prediction is the inversion of a permutation test, a specific +type of randomization test (see, e.g., Chernozhukov et al. (2018)). More generally, although it is unusual to +invert a randomization test, we show that by applying certain transformations to the data, our randomization +tests can be inverted to construct confidence intervals in semiparametric models (see, e.g., Rabideau and +Wang (2021)). Section 3.3 delineates precisely how our tests can be inverted to produce conformal/confidence +intervals. Due to this inverse relationship between prediction/estimation and testing, we generally (with the +exception of Section 3.3) restrict our attention solely to hypothesis testing for clarity of presentation, and +only state results in terms of the validity of such tests (and, in particular, the stochastic domination of the +uniform distribution by their p-values). +Finally, all of the problem domains which motivate this work are centered around adaptively collected data. +Indeed, as we show in this paper, we will be able to answer all of these inferential questions on adaptively +collected data via a general framework for randomization testing as long as the adaptivity is known to the +analyst. That is, we assume that the analyst has access to the (potentially non-deterministic) decision- +making algorithm used to assign treatments in the data. We now introduce some notation to define this +setup and formalize these notions. +1.2 +Notation +The adaptive data collection settings we consider in this paper—of which popular settings like bandits, +Markov Decision Processes (MDPs), reinforcement learning, and adaptive experimental designs are all special +cases—comprise a data collection horizon of T, the number of timesteps of data collected by the practitioner +(we will treat T as deterministic in this paper). At each timestep t, a context (sometimes called a state) +Ct ∈ C is observed, an action (also referred to as a treatment) Xt ∈ X is selected, and a response (called +an outcome or reward in some situations) Yt ∈ Y is subsequently observed. +Z := C × X × Y is the +(assumed time-invariant) sample space of the triple Zt := (Ct, Xt, Yt)—in this paper, we assume that Z +is discrete; see Remark 1 for an explanation of why as well as how to handle the case of continuous Z. +More formally, define the history at time t, Ht, to be the sequence ((C1, X1, Y1), . . . , (Ct, Xt, Yt)); H0 is +simply defined to be empty ∅. Then the action Xt is selected conditionally on Ht−1 and Ct. These action- +selection probabilities are encoded in the known decision-making (i.e., adaptive assignment) algorithm A, +where the probability of selecting action x ∈ X at the tth timestep given the tth context and prior history +3 + +of context-action-response triples up until time t is given by PA(x|Ct, Ht−1). The joint density of the full +data sequence HT = ((C1, X1, Y1), . . . , (CT , XT , YT )) is denoted by f. Unless otherwise noted, we always +consider Markovian systems, and thus assume that: +Yt ⊥⊥ Ht−1 | (Ct, Xt).1 +(1) +In this paper, we also sometimes consider domains without contexts, in which case Ct = ∅ for all t ∈ [T]. For +a given data sequence (C1, X1, Y1), . . . , (CT , XT , YT ), set D := HT to be their concatenation; take D := ZT +to be the sample space of the random variable D. Finally, as a note on general notation, we use [n] to refer +to the set {1, . . . , n} and write [n : m] to denote {n, . . . , m} for integers n ≤ m. +1.3 +Contribution +In this paper, we provide exactly valid randomization-based inference for adaptively collected data. We +make the following contributions, outlined below: +We derive a weighted Monte Carlo (MC) randomization testing framework. This test is able to resample +data from essentially any arbitrary resampling distribution (as long as a certain proportionality hypothesis +is satisfied) and, by weighting the samples according to certain likelihood ratios, produce a p-value. Along +the way, we show that we are also able to utilize an unweighted Markov Chain Monte Carlo (MCMC) +randomization procedure developed in Besag and Clifford (1989); Berrett et al. (2020); Bates et al. (2021); +Barber and Janson (2022+), and applied in other settings, to perform inferential tasks on adaptively collected +data. Both approaches offer exactly valid Type-I error control at essentially whatever computational cost +the user desires (although, more computation generally results in higher power). Our novel weighted MC +approach, however, is empirically no less powerful than the MCMC procedure and is, in fact, often more +powerful in our simulations, especially when the number of resamples is small. +We use both the weighted MC and unweighted MCMC frameworks to test against a number of general null +hypotheses on adaptively collected data: +1. Letting g be any function of X, we can test +Yt ⊥⊥ Xt | (Ct, g(Xt)) +(2) +in particular settings by resampling copies of Xt for each t ∈ [T] from any distribution conditional on +g(Xt) and weighting these resamples by certain likelihood-based ratios. In the special case in which g is +a constant function, Xt can sometimes be sampled in such a way such that no weighting is required. In +particular, the prior works of Pocock and Simon (1975); Simon (1979); Bojinov and Shephard (2019); +Ham and Qiu (2022) are all able to handle this special setting by simply resampling the Xt’s by +fixing the contexts and responses and re-running A. In general, however, when g is non-constant, this +unweighted procedure cannot be applied, yet our procedure always can be. One example of such a non- +constant g might map treatments x in a multidimensional treatment space X to their first dimension +x1. The hypothesis being tested in equation (2) would then be of conditional independence of the +response with all other dimensions of the treatment given the context and first treatment dimension. +Additionally, our framework applies in challenging settings like MDPs, whereas that of prior work does +not. Beyond this added generality of our testing procedure, it offers various improvements (in terms +of power and potential computational efficiency) over the existing work in certain settings. +2. We can test for non-stationarity in the conditional distributions Yt | Xt, Ct in various complex adaptive +data collection settings. +3. Under the semiparametric assumption that Yt | Xt, Ct follows a location model with location parameter +additive in Xt, our first test can be inverted to construct confidence regions for the location differences +1This assumption will only become relevant in Section 3. Additionally, while this Markovianity assumption is standard in +the adaptive data collection literature (e.g., Hahn et al., 2011; Zhang et al., 2020; Bibaut et al., 2021), we discuss in Remark 4 +how our procedure can be applied to test slightly different hypotheses when no Markovianity assumption is made. +4 + +between actions as well. Similarly, our non-stationarity test can be inverted to construct conformal +prediction regions for YT under any adaptive assignment algorithm, considerably generalizing existing +work which can only perform conformal inference under certain very specific adaptive assignment +algorithms involving only a single round of adaptivity (Tibshirani et al., 2020; Fannjiang et al., 2022) +or can only do so approximately (Chernozhukov et al., 2018; Gibbs and Cand`es, 2021; Barber et al., +2022). +Within our framework, it is more desirable (in terms of power) to have the weight of each resample as close +as possible to that of the observed data; developing resampling procedures which achieve this for the various +inferential tasks and environments discussed above is challenging. As such, we devise a number of novel and +computationally tractable resampling procedures to be used for our tests. +Finally, to demonstrate the practicality of our randomization tests (which can be deployed at essentially +any user-specified computational load) as well as the statistical efficiency of our resampling procedures, +we perform a simulation study. By considering the aforementioned inferential tasks in a variety of data- +generating environments, and by using data collected by a number of decision-making algorithms—including +both deterministic and randomized ones—we demonstrate that our resampling procedures produce a test +which is both statistically efficient (in terms of power and average confidence/prediction interval length) and +computationally tractable, while, of course, also guaranteeing exact validity. +1.4 +Related work +1.4.1 +Randomization tests for adaptively collected data +As noted in Rosenberger et al. (2019), the works of Pocock and Simon (1975) and Simon (1979) consider +randomization testing for adaptively collected data consisting of actions and outcomes in the Markovian +setting of equation (1) under a certain restricted set of assignment algorithms. Extending their approach, +Ham and Qiu (2022) are able to additionally handle context variables. Bojinov and Shephard (2019) also +develop a randomization test that generalizes that of Pocock and Simon (1975); Simon (1979) primarily by +relaxing the Markovianity assumption. The key to the approach put forth in all of the aforementioned papers +is that they restrict the adaptivity of the data collection environment and consider null hypotheses such that +simply fixing the contexts and responses and resampling the treatments via the known adaptive assignment +algorithm produces exactly exchangeable copies of the data, thereby enabling them to conduct a standard +unweighted randomization test. On the other hand, our main result, the weighted MC randomization test +(Theorem 2.1) makes no such assumptions about the data-generating distribution nor the resampling distri- +bution other than that they satisfy a certain proportionality hypothesis. Also, as we empirically demonstrate +in Section 5, our approach can be powerful even on data generated by deterministic assignment algorithms in +the Markovian setting, while that of Pocock and Simon (1975); Simon (1979); Bojinov and Shephard (2019); +Ham and Qiu (2022) are provably powerless. For more details regarding the comparison of our work to prior +work see Remarks 3 and 4. +Inference in bandits and reinforcement learning +Existing works for performing inference in rein- +forcement learning settings are typically either asymptotic (e.g., Lai and Wei, 1982; Deshpande et al., 2018; +Zhang et al., 2020; Hadad et al., 2021; Zhang et al., 2021, 2022) or, if not, are conservative in their finite- +sample bounds (e.g., Howard et al., 2021; Kaufmann and Koolen, 2021). As opposed to these works, all our +hypothesis tests are able to maintain exact and non-conservative validity in finite samples. +1.4.2 +MCMC and conditional permutation/randomization tests +The connection between sampling exchangeable samples and MCMC was first developed by Besag and +Clifford (1989) and has more recently been utilized by Berrett et al. (2020); Bates et al. (2021); Barber and +Janson (2022+), for the purposes of efficiently running a conditional permutation test, generating knockoffs, +and approximate co-sufficient sampling, respectively. The MCMC randomization test developed by Besag +5 + +and Clifford (1989) which we outline in Section 2 takes the same approach: it generates exchangeable samples +using base draws from a Metropolis–Hastings Markov Chain. However, to the best of our knowledge, our +work is the first that applies this MCMC approach to adaptively collected data. +1.4.3 +Conformal inference +Our weighted MC randomization test, when applied to non-stationarity testing and inverted, has close +connections with conformal inference. The works of Tibshirani et al. (2020) and Fannjiang et al. (2022) are +most related to our weighted MC randomization test when applied to non-stationarity testing and conformal +inference. Tibshirani et al. (2020) first extend conformal inference from i.i.d. data to the setting of covariate +shift, under the assumption that the likelihood ratio between test and train covariates is known by developing +a more general weighted conformal prediction (WCP) procedure. Relatedly, Hu and Lei (2020) apply WCP +to give an asymptotically powerful non-stationarity test in the same setting. The work of Fannjiang et al. +(2022) extends Tibshirani et al. (2020) and shows how to handle dependence between train and test sets in +the setting of feedback covariate shift. These prior methods handle only a single round of adaptivity (i.e., the +first T − 1 rounds of “train-time” data is fully i.i.d., but the “test-time” covariate distribution at timestep +T may either be different (Hu and Lei, 2020; Tibshirani et al., 2020) or depend on the first T − 1 datapoints +via feedback covariate shift (Fannjiang et al., 2022)). In contrast, our test for non-stationarity (the inversion +of which yields a conformal prediction interval) is able to handle any number of rounds of arbitrary analyst +adaptivity. +Another related line of work is in handling fully non-exchangeable data; that is, in contrast to Tibshirani +et al. (2020) and Fannjiang et al. (2022), no exchangeability is assumed within any particular train set. In +particular, Chernozhukov et al. (2018), Gibbs and Cand`es (2021), Xu and Xie (2021), Barber et al. (2022) +all consider various versions of non-exchangeable data and extend conformal inference to these regimes. The +methods presented in these works, however, are anti-conservative with bounds degrading as the degree of +non-exchangeability grows. In contrast, the methods presented in this paper (using the adaptivity known to +the analyst) provide exact validity in the adaptive (and hence potentially highly non-exchangeable) settings +in which they apply. +1.5 +Outline +In Section 2 we introduce the weighted MC randomization test. We also briefly describe how to perform an +unweighted randomization test by using MCMC sampling. We then, in Section 3, show how our approach +can be applied to solve a wide range of difficult inferential problems on adaptively collected data. Then in +Section 4 we introduce a number of novel algorithms for generating resamples that can be used for a variety +of inferential tasks and adaptive data collection environments. Finally, in Section 5, we perform a simulation +study of the methods introduced in Sections 2 and 3, using the resampling procedures from Section 4, that +both empirically validates our methods by illustrating their validity and computational tractability, and +demonstrates their power in a variety of challenging settings. +2 +The weighted MC randomization test +In this section we introduce our main approach to randomization testing: a novel weighted MC randomiza- +tion test. Additionally, at the end of this section, we briefly review an unweighted MCMC randomization +testing framework introduced by Besag and Clifford (1989), but, to the best of our knowledge, never before +applied to adaptively collected data (as we do in Section 3). The key to both approaches is the ability to +define likelihood-ratio-based weights that satisfy a certain proportionality hypothesis, which, as we show in +Section 3, is possible in a range of inferential tasks in various adaptive data collection settings. Empiri- +cally, our weighted MC approach never performs worse than the unweighted MCMC test, and indeed often +dominates it, especially when the number of resamples is small. +6 + +Our weighted MC randomization test is a randomized procedure, taking the data set D as input, which sam- +ples m conditionally i.i.d. MC draws ˜D(1), . . . , ˜D(m) given D. Define D to be the list ( ˜D(0), ˜D(1), . . . , ˜D(m)) +where ˜D(0) := D and let q( ˜D(i)|D) denote the conditional probability of sampling ˜D(i) from this condi- +tional distribution given that the dataset D was observed. Setting ˆf( ˜D(i)) := �T +t=1 PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t ), the +procedure then calculates likelihood-ratio-based weights for each resample ˜D(i) in D: +wq +D( ˜D(i)) = +ˆf( ˜D(i)) � +k∈[0:m]\{i} q( ˜D(k)| ˜D(i)) +�m +j=0 ˆf( ˜D(j)) � +k∈[0:m]\{j} q( ˜D(k)| ˜D(j)) +, +(3) +where ˜C(i) +t , ˜X(i) +t +and ˜H(i) +t +are, respectively, the context at, action at, and history up until timestep t of the +ith resample ˜D(i). Finally, the procedure outputs the p-value +p := +m +� +i=0 +wq +D( ˜D(i))1[S( ˜D(i)) ≥ S(D)], +(4) +where S : D → R is any test statistic. We summarize this procedure in the pseudocode of Algorithm 1 +Algorithm 1: Weighted Monte Carlo randomization test +Input: Data sequence D, resampling distribution ˜q, number of samples m, test statistic S +1 Sample ˜D(1), . . . , ˜D(m) | D +i.i.d. +∼ ˜q(·|D) +2 D ← ( ˜D(0), ˜D(1), . . . , ˜D(m)) where ˜D(0) := D +3 Compute wq +D( ˜D(i)) for each i ∈ [0 : m] via equation (3) +Output: p, calculated via equation (4) +The above weighting scheme may not always yield a valid p-value. We show, however, that as long as the +hypothesis +H∝ +0 : ˆf( ˜D(i)) = Kf( ˜D(i)), ∀i ∈ [0 : m] for some constant K not depending on i, +(5) +holds, then p is a valid p-value. Importantly, note that the left hand side of the proportionality (5) is always +computable (since it depends only on A, which is known), whereas the right hand side is not in general since +the density f is unknown. As we show in Section 3, H∝ +0 holds for a number of null hypotheses in various +adaptive data collection settings, thereby allowing for the valid use of the above to test against such nulls. +We now present the main result of this paper, which is that, under H∝ +0 , p is a valid p-value. +Theorem 2.1. The p-value defined in equation (4) stochastically dominates the uniform distribution under +equation (5) for any resampling distribution ˜q satisfying H∝ +0 . +While we relegate the proof of Theorem 2.1 to Appendix A, we note here that the proof is quite different +than those of randomization tests in the standard unweighted case. For example, the proof of validity of +the standard (unweighted) randomization test for exchangeable data and resamples—which is essentially an +immediate consequence of the joint exchangeability of the original data and all resamples (see, e.g., Besag +and Clifford (1989); Edgington and Onghena (2007); Candes et al. (2018); Berrett et al. (2020); Bates et al. +(2021))—clearly will not apply in our case, since the data and resamples are not jointly exchangeable and +the weighting scheme further makes it unclear how any such exchangeability could be used to prove validity +of p. A second, more recent proof technique, given by Hemerik and Goeman (2018) in the context of the +MC permutation test on exchangeable data, is to condition on the set orb(D) of all possible permutations of +the observed data and use the fact that, conditionally on orb(D), the original data is uniform over orb(D) +and the set of resampled permutations is also uniform to show that the test is valid. In our setting, however, +due to the non-exchangeability, this conditional distribution need not be uniform. +Rather, the proof of Theorem 2.1 is via the application of a simple yet new (to the best of our knowl- +edge) application of Bayes’ theorem and proceeds by showing the conditional validity of p given the list +D; see Appendix A for details. Furthermore, as discussed in Remark 8, the assumption that the resamples +˜D(1), . . . , ˜D(m) are drawn conditionally i.i.d. given D can be relaxed and a more general test (with generalized +weights) can be used; see Appendix E.1 for details. +7 + +Remark 1. The proof of Theorem 2.1 given in Appendix A applies only to discrete data distributions f +and resampling distributions q (recall we assumed all data was discrete in Section 1.2). +We focus only +on such distributions as in any real-world application of our test, the computer on which our test is being +run on has only finite precision, rendering both the original dataset as well as all resamples discrete. That +is, any practitioner running our test on a computer is necessarily dealing with discrete data, not due to +any restrictions of the test itself, but rather the finite precision limitation of the computer. +This is not +to say, however, that continuous distributions should never be considered in any situation. For example, +in an asymptotic theory, rates of convergence for discrete distributions may degrade as the resolution of +discretization becomes finer. In such a case, a continuous approximation of a computer-rendered discrete +distribution may be considerably more useful than directly handling the discrete distribution itself. The key +difference in our case, however, is that our theory is exact in finite samples for any discrete distribution. +Hence, the guarantee of Theorem 2.1 does not worsen at finer discrete resolutions, but rather remains intact, +thereby permitting our consideration of only discrete distributions. +We note, however, that our procedure should apply much more broadly to continuous distributions, but +perhaps not completely generally. See (Huang and Janson, 2020, Figure 5) for one such example in which +our theory does not hold because Bayes’ theorem is violated, by defining a dataset with T = 2 as well as a +resampling distribution with m = 1 for which the marginal distribution of the resample P( ˜D(1)) is not equal +to ˜q( ˜D(1)|D)f(D) by taking X1 ∼ Unif(0, 1) and defining ˜q to sample ( ˜X(1) +1 , ˜X(2) +1 ) ∼ LX1 where LX1 is the +segment of length 1 orthogonal to the x-axis, intersecting it at the point X1, with an angle of (1 − X1)10π/2 +with the y-axis. +Remark 2 (Randomizing for exact Type-I error control). Just as with a standard randomization test and, +as a special case, conformal inference in which the procedure is called smoothing (e.g., Vovk et al., 2005), +one can define a “lower” p-value +p− := +m +� +i=0 +wq +D( ˜D(i))1[S( ˜D(i)) > S(D)] +and randomize to obtain a test for Theorem 2.1 which controls Type-I error at exactly the nominal level. +Specifically, the test which fails to reject when p− > α, rejects with probability α−p− +p−p− when p− ≤ α < p, and +otherwise always rejects, controls Type-I error at exactly the nominal level α. +While Theorem 2.1 gives one great freedom and generality in computing valid randomization p-values by +allowing for many choices of the conditional distribution q, two aspects in particular remain, at present, +quite unclear: (a) in what situations and under what null hypotheses can we ensure that the proportionality +hypothesis H∝ +0 holds, as well as (b) which choices of q one should sample from to obtain a powerful test. We +address the first question in the context of adaptively collected data in Section 3 and discuss the second in +Sections 4 and 5. +The unweighted MCMC randomization test +Above, we discussed a weighted MC randomization +test which weights resamples based on certain likelihood ratios. +Here, we briefly outline an unweighted +MCMC randomization test developed by Besag and Clifford (1989). While the test itself is not new (see, +e.g., Berrett et al. (2019); Bates et al. (2021); Barber and Janson (2022+) for more recent utilizations and +extensions of the test), our application of it to adaptively collected data, to the best of our knowledge, is. The +randomized procedure for the unweighted MCMC randomization test repeatedly takes Metropolis–Hastings +steps, starting at D by, at each step i, sampling a proposal ˜D given ˜D(i−1) from q(·| ˜D(i−1)), and additionally +computing an acceptance ratio under the stationary distribution f to decide if the Metropolis–Hastings +step accepts the proposal or remains at ˜D(i−1). Just as with Algorithm 1, the acceptance ratio calculation +actually only involves known quantities depending on ˆf and q and hence the validity of the MCMC test is +again implied by the proportionality hypothesis H∝ +0 . See Besag and Clifford (1989) for a delineation of the +general test. We also note here that, in all of our simulations, our weighted MC test performs no worse +than—and often dominates—the unweighted MCMC test. As such, exploring the utility of our weighted MC +test in the settings of Berrett et al. (2020); Bates et al. (2021); Barber and Janson (2022+), all of which use +the unweighted MCMC procedure, may be of interest for future work. +8 + +3 +Randomization testing for adaptively collected data +In this section, we apply the randomization tests from Section 2 to solve a host of challenging inferential tasks +in adaptive data collection settings. In Section 2, we saw that to perform a weighted MC randomization +test we needed to be able to run Algorithm 1 which in turn required us to choose a q that satisfies the +hypothesis H∝ +0 . In this section, we show that, for a number of null hypotheses in various adaptive data +collection settings, H∝ +0 can be satisfied with many non-trivial choices of q, thus allowing for the application +of the weighted MC randomization test. We note here that, since the focus of this section is ensuring that +H∝ +0 holds (under various null hypotheses and adaptive data collection settings), all results also immediately +apply to the unweighted MCMC test. +We address two broad classes of tasks and describe each—as well as the adaptive data collection environments +in which we consider them—in detail in the sections that follow: +1. Testing conditional independence between treatment and response conditional on context and some +function of the treatment (Section 3.1). Applying certain transformations to the data and inverting +such tests allows us to construct confidence intervals for response distribution parameters in certain +semiparametric models (Section 3.3.1). +2. Non-stationarity testing (Section 3.2) and its application to conformal inference (Section 3.3.2) +In Sections 3.1 and 3.2 below, we discuss a number of adaptive data collection environment setups. Each of +these setups assumes some combination/subset of equations (1) (Markovianity), (6), (7a), (7b): +Yt | Ct, Xt does not depend on t +(Y -Stationarity) +(6) +This first assumption asserts that the environment is Y -stationary so that the conditional response distribu- +tion does not vary with time; this distinction will be important in whether or not additional randomization +can be employed for a more powerful conditional independence test. +Ct ⊥⊥ (X1, . . . , Xt−1) | (C1, Y1, . . . , Ct−1, Yt−1) +(Weak non-reactivity) +(7a) +Ct | Ht−1 depends neither on t nor Ht−1 +(C-stationarity & strong non-reactivity) +(7b) +The assumptions of equations (7a) and (7b) are related and describe the environment through the behavior +of contexts. In particular, the weak non-reactivity assumption (equation (7a)) says that the actions prior +to time t do not affect Ct, conditionally on the previous contexts and responses; i.e., the future states +of the environment are essentially (conditionally) non-reactive to prior actions. This weak non-reactivity +assumption is also made in (Ham and Qiu, 2022, Assumption 1). The C-stationarity & strong non-reactivity +assumption (equation (7b)) is a stricter version of this, and stipulates that each context is generated by the +environment independently from the past and also that the distribution from which it is generated does not +change with t. +Before proceeding, we pause here to emphasize that all results that ensue in this section hold for any adaptive +assignment algorithm A, as long as it is known. We present all results by first describing the null hypothesis +being tested, then explaining the various environments in which it can be tested, and finally delineating any +constraints on the resampling distribution q in each of these environments so as to ensure that H∝ +0 holds +under the null. +3.1 +Conditional independence testing +Null hypothesis +Let g be any function of x ∈ X, with unspecified codomain. We wish to perform a +conditional independence test between treatment and response given both context and the g-evaluation of +9 + +the treatment: +H⊥⊥,g +0 +: Yt ⊥⊥ Xt | (Ct, g(Xt)), ∀t ∈ [T]. +One example in which this hypothesis might be employed is in the problem of A/B testing in online advertis- +ing, where a company presents users with the same ad but with differing hues, saturations, and brightnesses +(i.e., so the treatment space is 3-dimensional). One concrete hypothesis that may be tested in this situation +is if saturation and brightness are enough to predict user response alone (i.e., is user response independent +of hue given both saturation and brightness?). Setting g(Xt) = (Xt,saturation, Xt,brightness) to map Xt to its +saturation and brightness components, H⊥⊥,g +0 +is equivalent to this hypothesis. A second scenario is in testing +if a particular subset of treatments induces the same response distribution. That is, letting X = {0, 1, 2} +be the treatment space, the null hypothesis that Yt ⊥⊥ Xt | (Ct, Xt ∈ {0, 1}) is equivalent to H⊥⊥,g +0 +with +g(Xt) = I(Xt = 2). Finally, in the special case in which g is a constant function, the hypothesis being tested +is that of simple conditional independence between treatment and response given context, and, for this par- +ticular choice of g, the unweighted test of prior work (Pocock and Simon, 1975; Simon, 1979; Bojinov and +Shephard, 2019; Ham and Qiu, 2022) can be employed in Environments 1 and 3 (but not in Environments 2 +or 4); as we describe below, however, our framework allows for a significant improvement in power over this +prior work when employed in Environment 3. +Environment 1 (Non-reactive environment). The non-reactive environment is one in which the Marko- +vianity (equation (1)) and weak non-reactivity (equation (7a)) assumptions hold. Examples of a non-reactive +environment include (contextual) bandits—a setting that is common in the mobile health literature (e.g., +Tewari and Murphy, 2017)—as well as many common adaptive experimental designs, such as adaptive crop +yield experiments (e.g., Alesso et al., 2021). +Environment 2 (Markov Decision Process). The next environment we consider is an MDP, which assumes +only the Markovianity assumption (equation (1)) holds, and also stipulates that Yt := Ct+1; note that this +implies that Assumption (7a) also holds. One example of MDP data arises in electronic health records (Li +et al., 2022). +Environments 3 and 4 below are Y -stationary counterparts of Environments 1 and 2, respectively. +Environment 3 (Stationary non-reactive environment). The most restricted of the environments we con- +sider, the stationary non-reactive environment, assumes Markovianity (equation (1)), Y -stationarity (equa- +tion (6)), and C-stationarity & strong non-reactivity (equation (7b)). Examples of this environment arise in +reinforcement learning as a stationary (contextual) bandit as well as in many common adaptive experimental +designs—one example being in adaptive clinical trials (e.g., Giles et al., 2003).2 +Environment 4 (Stationary MDP). With the additional assumption of stationarity, the stationary MDP +is a special type of MDP in which Y -stationarity (equation (6)) also holds, thereby assuming that the MDP’s +transition distribution does not change across timesteps. An example of stationary MDP data is in patient +admissions to the emergency room during surge demand (e.g., Lee and Lee, 2018).3 A second example arises +in robotics, where the robot’s dynamics and interactions with its environment can be modeled as a stationary +MDP (e.g., Su´arez-Hern´andez et al., 2019). +Constraints on q +The resampling procedure q may incorporate two sources of randomization: (a) ran- +domizing the order of the data by some permutation in a certain environment-specific set Π, and (b) ran- +domizing the actions conditional on their g-evaluations. More specifically, q must be such that the sequences +of contexts, responses, and g-evaluations of the treatment in each resample are equal to some (random) +2To distinguish this setting from the adaptive crop yield experiment example, note that in an adaptive clinical trial, patients +are selected i.i.d. from some population, thus obeying the C-stationarity & strong non-reactivity (equation (7b)). On the other +hand, in an adaptive crop yield experiment, due to weather fluctuations (such as temperature and precipitation) over time, this +assumption is violated. For the same reason, we expect adaptive clinical trial settings, but not adaptive crop yield experiments, +to be Y -stationary. +3Again, this example differs from that of electronic health records in Environment 2 because here, we do not expect the +transition dynamics to change over time, whereas in the previous example, administering certain medication to a patient may +in fact change their internal state and thus alter the way in which they react to the same medication later on. +10 + +permutation—albeit, the same one—π ∈ Π of their respective sequences in the original data: +� +˜C(i) +t , g( ˜X(i) +t ), ˜Y (i) +t +� += +� +Cπi(t), g(Xπi(t)), Yπi(t) +� +, ∀t ∈ [T], for some πi ∈ Π. +(8) +Note that equation (8) does not force ˜X(i) +t += Xπi(t), and thus allows the treatments to be further randomized +over X, so long as the restriction g(Xπi(t)) = g( ˜X(i) +t ) is met. Finally, the restricted set Π of allowable +permutations is environment-specific; we first focus on Environments 1–3, where the non-stationarity of +Environments 1 and 2 prevent us from permuting the data at all, while the stationarity of Environment 3 +allows for any permutation: +Proposition 3.1. Let Πi, i ∈ [3] denote the set of allowable permutations for Environments 1-3, above. Then +if Π1 = Π2 = {id}, where id denotes the identity permutation, and Π3 = Π[T ], the set of all permutations on +[T], the proportionality hypothesis H∝ +0 is satisfied under H⊥⊥,g +0 +. +Proof. In Environments 1 and 2, the restriction that Π1 = Π2 = {id} ensures that +ˆf( ˜D(i)) = +T +� +t=1 +PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) += +�T +t=1 Pt(Yt|Ct, g(Xt), ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P(Ct|C1, Y1, . . . , Ct−1, Yt−1) +�T +t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, . . . , Ct−1, Yt−1) +by H⊥⊥,g +0 +and that g( ˜X(i) +t ) = g(Xt) += +�T +t=1 Pt( ˜Y (i) +t +| ˜C(i) +t , ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P( ˜C(i) +t | ˜C(i) +1 , ˜Y (i) +1 , . . . , ˜C(i) +t−1, ˜Y (i) +t−1) +�T +t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, . . . , Ct−1, Yt−1) +as Π1 = Π2 = {id} & equation (8) += +1 +�T +t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, . . . , Ct−1, Yt−1) +f( ˜D(i)), +due to equation (7a) +where the subscript t on P is used to emphasize that the conditional distribution Yt | Ct, Xt may depend on +t. Hence, the proportionality hypothesis is satisfied with K = +1 +�T +t=1 Pt(Yt|Ct,g(Xt),Xt)P(Ct|C1,Y1,...,Ct−1,Yt−1). +On the other hand, in Environment 3, Π3 = Π[T ] and any permutation is allowed since +ˆf( ˜D(i)) = +T +� +t=1 +PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) += +�T +t=1 P(Yπi(t)|Cπi(t), Xπi(t))PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P(Cπi(t)) +�T +t=1 P(Yt|Ct, Xt)P(Ct) +due to equations (6) and (7b) += +�T +t=1 P(Yπi(t)|Cπi(t), ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P(Cπi(t)) +�T +t=1 P(Yt|Ct, Xt)P(Ct) +by H⊥⊥,g +0 +and that g( ˜X(i) +t ) = g(Xπi(t)) += +�T +t=1 P( ˜Y (i) +t +| ˜C(i) +t , ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P( ˜C(i) +t ) +�T +t=1 P(Yt|Ct, Xt)P(Ct) +by equation (8) += +1 +�T +t=1 P(Yt|Ct, Xt)P(Ct) +· f( ˜D(i)), +due to equation (7b) +where we have dropped the subscript t on P used above (and also do not need to subscript P(Cπi(t)) due to +the Y -stationarity assumption of equation (6) (resp. the C-stationarity assumption of equation (7b)). Hence, +once again, the proportionality hypothesis holds, but now with K = +1 +�T +t=1 P(Yt|Ct,Xt)P(Ct). +The precise definition of the allowable set of permutations Π4 in Environment 4 is somewhat more complicated +due to the serial dependence between consecutive timesteps (i.e., Ct+1 is generated conditionally on (Ct, Xt)) +11 + +that is not present in Environment 3. In particular, only permutations which preserve the property that the +tth response is equal to the (t + 1)th context are allowed. We formally state a Proposition here, but relegate +its proof to Appendix B: +Proposition 3.2. Setting +Π4 := {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1}, +the proportionality hypothesis H∝ +0 is satisfied under H⊥⊥,g +0 +in Environment 4. +We briefly note here that unless the MDP’s state space C is relatively small, then the (data-dependent) +permutation set Π4 will typically be quite small and virtually no randomization in permutations can be +performed. There are, however, settings in which the state space C is small, such as in the restless multi- +armed bandit considered by Mate et al. (2020, 2021) in the context of patient adherence monitoring and +well-being, in which our test can be used effectively. +Remark 3 (Comparison with existing work). In Environment 1 (and Environment 3, which is a special +case), when g is a constant function, the unweighted randomization test of Pocock and Simon (1975); Simon +(1979); Bojinov and Shephard (2019); Ham and Qiu (2022) applies. However, the testing procedure presented +above has two advantages over those in prior work. First, by randomizing timesteps in addition to treat- +ments, our procedure is more powerful than theirs in the stationary setting of Environment 3, especially for +deterministic assignment algorithms for which their procedure is powerless; we empirically demonstrate this +in Section 5.1.1. Second, their procedure requires that each resample rerun A independently m times with the +same sequence of Ct and Yt as in D.4 While this would seem to be the most natural and statistically powerful +approach, it may not always be computationally feasible. In such a case, our method, by using a more easily +computable resampling procedure, provides a computationally tractable workaround (albeit perhaps at the cost +of degraded statistical efficiency per resample). As a concrete example, the adaptive assignment algorithm A +used to generate the original data could be based on Thompson sampling in a complex Bayesian model, thus +rendering it too computationally burdensome to run for any but a very small number of MC samples. On +the other hand, unnormalized densities—which are all that our procedure requires, due to the proportionality +assumption H∝ +0 —are generally easy to compute, thus rendering computation of the weights in equation (3) +tractable under a less computationally intensive resampling procedure q. +Remark 4 (Relaxing structural assumptions). In Appendix C, we show the above testing procedure can +be generalized to arbitrary adaptive data collection processes in which none of the assumptions of equations +(6),(7a),(7b) nor the Markovianity assumption of equation (1) are made, and the adaptive data collection +environment is assumed to be completely general. In such an environment, we show that one can, somewhat +analogously, consider a sequence of functions g1, . . . , gT and test simultaneous (over t) conditional indepen- +dence between the tth context (as well as the tth response) and the prior sequence of actions given the prior +sequences of contexts and responses as well as the sequence comprising the gs-evaluation of the sth action +for s ∈ [t]. The special case of this hypothesis wherein all the gt are constant allows for unweighted random- +ization testing in the completely general environment described in this Remark as shown by the prior work +of Bojinov and Shephard (2019). +3.2 +Testing for non-stationarity +Null hypothesis. For our non-stationarity test, the null hypothesis HS +0 is that the response distribution is +stationary (but unknown). That is, the conditional distribution of response given the context and treatment +is the same across all timesteps: +HS +0 : Yt | (Ct, Xt) does not depend on t. +In this section, we consider two environments. The first is a a special type of non-reactive environment +(Environment 1): +4Pocock and Simon (1975) and Simon (1979) only describe how to do so for certain adaptive assignment algorithms, and +both, as well as Bojinov and Shephard (2019), only consider the non-contextual case. +12 + +Environment 5 (C-stationary strongly non-reactive environment). The C-stationary strongly non-reactive +environment is an environment in which equation (7b) holds, in addition to the Markovianity assumption +of equation (1). +Once again, (contextual) bandits and various adaptive experimental designs are special +cases. One concrete example is in adaptive experimental designs studying the effects of job search assistance +programs on helping job seekers find work (Caria et al., 2020) over short periods of time since, in these +brief time intervals, we expect the “context” surrounding each individual (e.g., background, credentials, etc.) +to be roughly i.i.d. One additional example of a C-stationary strongly non-reactive environment for which +our theory also holds is an episodic MDP, in which each episode is viewed as a single time step, the reward +sequence a single response, and the action sequence a single (high-dimensional) action.5 +The second environment we consider is simply the MDP of Environment 2. +Constraints on q +By a proof similar to that of Proposition 3.1 for Environment 3, it is straightforward +to see that, as long as each draw from the resampling distribution q is (any) permutation of the original +data D, the proportionality hypothesis H∝ +0 holds under HS +0 in Environment 5. That is, as long as we set +Π = Π[T ] and do not allow for any other randomization, then ˆf( ˜D(i)) ∝ f( ˜D(i)). Under the MDP setting of +Environment 2, we may use the same randomization set of permutations Π4 stated in Proposition 3.2 and +again, do not include any additional randomization. We summarize this below: +Proposition 3.3. If Π = {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1} in Environment 2 and +Π = Π[T ] in Environment 5, and the only randomization in q’s resampling consists solely of permutations +drawn from Π, then the proportionality hypothesis H∝ +0 is satisfied under HS +0. +3.3 +Inverting tests to construct confidence and prediction intervals +3.3.1 +Confidence regions in semiparametric models +Recall that the hypothesis tests in Section 3.1 were all nonparametric and focused on testing (conditional) +independence and distributional equality. +In this section, we turn to the problem of exact parametric +inference in semiparametric models—focusing on semiparametric location models as a case study—and use a +technique previously used in standard randomization testing to construct confidence intervals (e.g., Rabideau +and Wang, 2021). Before proceeding, we emphasize that parametric inference for adaptively collected data +is a challenging problem and, as far as we are aware, all examples in the literature are either asymptotic +(e.g., Deshpande et al., 2018; Zhang et al., 2020; Hadad et al., 2021), or conservative (see, e.g., Howard et al. +(2021); Kaufmann and Koolen (2021)). +Now, consider the setting in which the response distribution is distributed according to a location family +with locations determined by action. More precisely, letting h0 denote a (unknown) base density, we assume +that at each time-step t ∈ [T], Yt | (Xt = x) ∼ h0(y − θx), where x �→ θx is some mapping of actions +to location parameters. In such a setting it is quite natural to ask: how much better or worse is action +x compared to x′, in terms of their location parameters? More formally, how can we test against the null +hypothesis HLoc,δ,x,x′ +0 +that θx′ − θx = δ for actions x, x′ ∈ X and δ ∈ R? +To test against HLoc,δ,x,x′ +0 +, we modify the dataset D and then perform a conditional independence test. +Specifically, by modifying the dataset by replacing Yt with Yt + δ · 1[Xt = x], we get that HLoc,δ,x,x′ +0 +implies +that x and x′ induce the same distribution over rewards in the modified data generating distribution. Hence +a test against H⊥⊥,g with g(Xt) = +� +{x, x′} if Xt ∈ {x, x′} +Xt otherwise +on this modified dataset serves as a test against +HLoc,δ,x,x′ +0 +. These same ideas can also be applied to scale families; see Appendix D for details. +Most importantly, these tests can all be inverted to construct confidence regions for the parameter in question, +namely θx′ − θx. For example, consider constructing a confidence region for the difference in locations of +5This episodic MDP setting also falls under the category of a (stationary) non-reactive environment, and hence our tests of +conditional independence presented in Section 3.1 also apply. +13 + +treatments x and x′ at nominal miscoverage rate α. One can construct the acceptance region by simply +running the test against HLoc,δ,x,x′ +0 +at level α at each δ in the parameter space ∆: those δ for which the test +fails to reject make up the acceptance region. In cases where the parameter space ∆ is either continuous or +too large to iterate over completely, approximate confidence regions can be constructed as follows: (a) grid +the space into a small finite set of discrete points ∆′ ⊆ ∆, (b) run the above procedure at each δ ∈ ∆′ to +obtain an accepted set of grid points A′, and (c) include all δ ∈ ∆ within a certain user-specified distance of +any of the accepted grid points of A′ in the confidence region. +3.3.2 +Prediction regions for YT +Similar to the previous section, the tests of Section 3.2 can be applied to construct prediction intervals for YT +before it is observed. In particular, again, one can construct the acceptance region for YT by running the non- +stationarity tests described in Section 3.2 on the almost-fully realized dataset ((C1, X1, Y1), . . . , (CT , XT , y)), +with y ranging over Y; just as with confidence intervals, in the case of large Y, the gridding/rounding +procedure described at the end of the previous section can be used to construct approximate prediction +intervals by simply selecting a small discrete set Y′ ⊆ Y. We do note, however, that it is not uncommon +for Y to be small or even categorical in many (challenging) settings and thus, for such problems, we can +grid Y directly to obtain exactly valid prediction regions. We now make a few remarks about some broader +connections of our procedure when used in this fashion to conformal inference, as well as challenges and +speedups in constructing intervals. +Remark 5 (Conditional conformal inference). While conditional conformal inference is impossible in general, +even in the case of i.i.d. data (Lei and Wasserman, 2014), Theorem 2.1 admits a conditional version which +may be useful when the covariate space X is finite and small. In particular, the validity of the test still +holds when we replace f with the conditional density f(·|XT ), which conditions on the “test-time” covariate +XT = xT ; thus, when inverting the hypothesis test and constructing a conformal prediction interval, one +can guarantee valid coverage conditional on XT (i.e., P(YT ∈ Cα +T (XT )|XT ) ≥ 1 − α, where Cα +m(XT ) denotes +conformal band at XT with nominal miscoverage rate α). In practice, this is possible as long as each resample +˜D(i) has ˜X(i) +T += XT (in addition to ˜D(i) being an allowable permutation of D), so that +f( ˜D(i)|XT ) � +k∈[0:m]\{i} q( ˜D(k)| ˜D(i)) +�m +j=0 f( ˜D(j)|XT ) � +k∈[0:m]\{i} q( ˜D(k)| ˜D(j)) += +f( ˜D(i)) � +k∈[0:m]\{i} q( ˜D(k)| ˜D(i)) +�m +j=0 f( ˜D(j)) � +k∈[0:m]\{i} q( ˜D(k)| ˜D(j)) +. +In turn, p’s validity once again ensues so long as the proportionality hypothesis H∝ +0 holds. +Remark 6 (Challenges with explicit construction of conformal bands). Split conformal inference (Pa- +padopoulos et al., 2007) is a variant of conformal inference which involves data splitting in order to construct +a conformal prediction region which can be explicitly and efficiently computed. Unfortunately, such an ex- +plicit construction of a conformal band using data splitting does not carry over using our procedure to test +non-stationarity, since the weights, through ˆf( ˜D(i)), may depend on the test-time response grid values y. Sim- +ilarly, the discretization procedure of Chen et al. (2018) used to construct explicit bands in the i.i.d. setting +by rounding the response to a small discrete set is not valid in our framework since, in general, probabilities +under PA involving the rounded data are either unknown or not efficiently computable. +Remark 7 (Sharing samples). To address the challenges of Remark 6, one can grid the space into a small +finite set Y′ and run the test at each y ∈ Y′, as discussed above in the case of continuous or large Y. Naively, +however, this involves drawing m resamples for each y ∈ Y′. To reduce the total number of resamples needed, +we can however share resamples between different values of y. That is, for y1, y2 ∈ Y′, resamples drawn in +association with y1 can be used to determine whether or not to accept y2 and vice versa, thereby more +effectively using each resample drawn. See Appendix E.2 for details. +14 + +4 +Resampling procedures +In Section 3, we focused on an information-theoretic question: in what data regimes can we define a sampling +procedure q for which our testing procedure is valid? Here, we turn to the second key question posed at +the end of Section 2: which choices of q are statistically best? Thus, while the last section specified how +to choose the the support of q (i.e., which variables should be randomized over and which should be held +fixed) depending on the null hypothesis being tested against and the adaptive data collection environment +in which it is being tested, this section explores what distributions to choose over these supports. In the +remainder of this section, we provide a partial answer to this challenging question by proposing a number +of resampling procedures that are compatible with the constraints outlined in Section 3 and will be shown +to be powerful and produce short confidence/prediction regions in Section 5. +Recall that we are focusing solely on conditionally i.i.d. resampling from procedures q and thus all re- +sampling/proposal distributions considered in this section can be applied to both the weighted MC and +unweighted MCMC tests (with q as the proposal distribution for the MCMC test). However, as mentioned +in Section 2, our weighted MC procedure does not in general require conditionally i.i.d. resampling and hence +the following remark describing how our test can be generalized in this case is in order: +Remark 8 (Non-i.i.d. resampling). When the resamples ˜D(1), . . . , ˜D(m) are not generated in a conditionally +i.i.d. manner, the test in Algorithm 1 can be slightly generalized. +In particular, letting Σ be any subset +of Π[0:m], the set of permutations on [0 : m], and ˜q(( ˜D(1), . . . , ˜D(m))|D) denote the conditional probability +of sampling ( ˜D(1), . . . , ˜D(m)) given that D was observed, the procedure can be generalized by redefining the +weights to be +w˜q,Σ +D ( ˜D(i)) = +ˆf( ˜D(i)) � +π∈Σ:π(0)=i ˜q(( ˜D(π(1)), . . . , ˜D(π(m)))| ˜D(i)) +�m +j=0 ˆf( ˜D(j)) � +π′∈Σ:π′(0)=j ˜q(( ˜D(π′(1)), . . . , ˜D(π′(m)))| ˜D(j)) +. +We relegate a description of this generalized procedure, as well as the proof of its validity, to Appendix E.1. +Remark 9 (Computational issues with conditionally i.i.d. resampling). We also note here that even with +a conditionally i.i.d. resampling scheme q, the computation of the p-value p can sometimes take Ω(m2) +computations. This is because, even with a conditionally i.i.d. resampling procedure, all pairs of conditional +densities q( ˜D(i)| ˜D(j)) must be computed and incorporated into the p-value computation. On the other hand, +if the conditional density draws samples ˜D(1), . . . , ˜D(m) such that q(·| ˜D(i)) is the same for all i ∈ [0 : m], +then +� +k∈[0:m]\{i} +q( ˜D(k)| ˜D(i)) = +�m +k=0 q( ˜D(k)| ˜D(i)) +q( ˜D(i)| ˜D(i)) +∝ (q( ˜D(i)| ˜D(i)))−1 +(9) +and so the p-value can be computed much more quickly in only O(m) computations by replacing � +k∈[0:m]\{i} q( ˜D(k)| ˜D(i)) +with (q( ˜D(i)| ˜D(i)))−1 in the weight equation (3); as a result, we focus on resampling procedures that have +this property. +The setup in this section is that we first consider resampling procedures used to test non-stationarity. We +then go on to discuss resampling algorithms for testing the conditional independence null hypothesis H⊥⊥,g +0 +, +some of which use some of the non-stationarity testing resampling algorithms as subprocedures in their +sampling process. All resampling procedures presented in this section are evaluated in our simulations in +Section 5. Additionally, more detailed pseudocode outlines of the resampling procedures in this section can +be found in Appendix F. Lastly, we make a brief note about the computation of probabilities under the +resampling distribution q. +Remark 10. All resampling procedures we consider in this section—and in our simulations in Section 5— +either sample uniformly or involve some sort of sequential sampling procedure. In either case, computation +of conditional probabilities under q are tractable as they are constant in the former and, in the latter, can be +calculated as the sample is generated by serially multiplying together the corresponding conditional probabil- +ities of each sequential sample as it is generated. Of course, probabilities of the form q(D| ˜D(i)) for i ∈ [m] +15 + +still must be computed (as the original dataset D is never resampled from q) from scratch, but this is simply +done in the same manner as above: behaving as though D had indeed been sampled from q conditionally on +˜D(i) and sequentially multiplying conditional probabilities of each resampleed timestep. +4.1 +Non-stationarity testing in a C-stationary strongly non-reactive environ- +ment +We first describe four types of resampling procedures that can be used for non-stationarity testing in a +C-stationary strongly non-reactive environment (i.e., Environment 5). As discussed in Section 3.2, all such +distributions must only randomize timesteps by permuting the data sequence. Apart from uniform permu- +tations, the other three procedures discussed in this section randomly permute the data in a way intended +to mimic A while also ensuring diverse random samples. We thus call these three resampling procedures +imitationπ, re-imitationπ, and cond-imitationπ, the common word imitation referencing the mimicking of A +(the prefixes re and cond will be explained later on in this section). All three procedures randomly permute +the data by serially sampling not-yet-sampled timesteps from the original data sequence. +uniformπ sampling +The uniformπ sampling procedure simply selects a permutation of the data uniformly +at random. Although simple, intuitively it may result in a diverse set of resamples. +imitationπ sampling +This sampling procedure samples permutations by sequentially resampling timesteps +from the original data, where the sampling distribution at time t acts as though the first t − 1 already- +resampled timesteps were drawn according to the true data generating-process and selects the tth timestep +proportionally to the probabilities dictated by PA. In other words, at each timestep t, letting R denote the +set of not-yet-sampled timesteps, the imitationπ distribution draws a timestep t′ from R, where the proba- +bility of drawing any given s ∈ R is proportional to PA(Xs|Cs, ˜Ht−1); if PA(Xs|Cs, ˜Ht−1) = 0 for all s ∈ R, +the procedure is ended and an attempt at a new resample can be begun using the same process. Finally, +(Ct′, Xt′, Yt′) is appended to ˜Ht−1 and t′ is removed from R. Intuitively, the imitationπ distribution behaves +as A would, feeding in the already-realized sequence of responses (Y1, . . . , YT ), except that it may only sample +amongst actions which correspond to not-yet-selected timesteps. See Algorithm 3 for pseudocode. +The re-imitationπ and cond-imitationπ distributions which we describe below are intended only for random- +ized decision-making algorithms A. When applied to a deterministic algorithm, they are both the same as +imitationπ. +re-imitationπ sampling +This distribution is similar to the imitationπ distribution, except that, to incor- +porate more diversity, it independently regenerates the exogenous randomness that the randomized decision- +making algorithm A makes as it mimics it to draw resamples; it thus rerandomizes the randomness of A. +More specifically, the re-imitationπ distribution views A as a sequence of decision rules δt which take as +input the tuple (Ct, Ht−1, U1, . . . , Ut), where Ut is the exogenous random variable generated by A at time t, +and output which action to take: +1. Sample a permutation of D by sequentially resampling timesteps from the data, where the sampling +distribution at time t uses the t − 1 already-resampled timesteps as well as the resampled exogenous +randomness ˜U1, . . . , ˜Ut−1, and then generates the random variable ˜Ut from Ut’s distribution, but con- +ditional on the remaining timesteps. Specifically, ˜Ut is sampled from the conditional distribution of Ut +given that +Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut−1, Ut) for at least one not-yet-selected timestep s. +If, however, this conditional distribution is degenerate (i.e., because there does not exist ˜Ut for which +Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut−1, ˜Ut) for any remaining timesteps s), the process is terminated and +sampling for a new resample is begun. +2. Select the tth timestep uniformly over all those not-yet-sampled triples (Cs, Xs, Ys) with +Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut). +(10) +16 + +The motivation behind the re-imitationπ distribution, as opposed to imitationπ, is that, by incorporating +more of the randomness used in the decision-making algorithm one may be able to obtain more diverse +samples while also better imitating the decision-making mechanisms of A. See Algorithm 4 for pseudocode. +cond-imitationπ sampling +The cond-imitationπ distribution is the same as re-imitationπ except that +instead of resampling the ˜Ut’s, it conditions on them and thus uses the same sequence of exogenous random- +ness as re-imitationπ does; this, of course, requires knowing the original sequence U1, . . . , Ut. Intuitively, this +conditioning that cond-imitationπ sampling performs should bias the weights closer to (m + 1)−1, resulting +in more powerful resampling. See Algorithm 5 for pseudocode. +4.2 +Non-stationarity testing in an MDP +We finally briefly discuss the resampling procedures for non-stationarity testing in an MDP (Environment 2). +We consider essentially the same four types of resampling procedures used for non-stationarity testing in a +C-stationary strongly non-reactive environment described in the last section. The only difference however, +is that, as discussed in Section 3.2, only a subset of permutations are allowed in the MDP setting so as to +ensure that the ˜Y (i) +t += ˜C(i) +t+1 condition that holds for i = 0 (i.e., in the observed data) also holds for each +resample ˜D(i). Thus, while the four procedures which we consider here—also called uniformπ, imitationπ, +re-imitationπ, and cond-imitationπ—sample timesteps serially without replacement according to A as their +analogs did in the previous section, they do so over only a restricted subset of not-already-sampled timesteps +at each round.6 +For this reason, the MDP data that we consider includes an additional action XT +1; hence the permutations +which we consider can be viewed as permuting the T +1 state-action pairs in this augmented dataset.7 More +specifically, suppose that at the end of round t − 1, the state-action pair (Cs′, Xs′) had just been selected +and appended to ˜Ht−2. Then, the set of allowable timesteps which can be sampled at round t are only +those not-yet-sampled state-action pairs (Cs, Xs) for which the state-action-next state triple (Cs′, Xs′, Cs) +is present in the original data sequence D. The precise way in which the timestep is sampled is then in +exact accordance with the weighting, randomization, and conditioning that correspond to the uniformπ, +imitationπ, re-imitationπ, and cond-imitationπ sampling procedures described in the previous section. See +Algorithms 6, 7, 8, and 9 for pseudocode. +4.3 +Conditional independence testing +We now describe four types of resampling procedures that can be used in a stationary non-reactive environ- +ment (Environment 3). Three of these resampling procedures randomize both timesteps and the action Xt +conditional on g(Xt) allowed by the stationarity, as discussed in Section 3.1. The procedure which does not +both randomize timesteps and actions simply randomizes the actions alone and is called imitationX; as such, +it can also be applied to the non-reactive environment (Environment 1) as well as an MDP (Environment 2). +Of the three procedures which randomize both timesteps and actions, two randomize the timesteps and +actions in two separate stages and therefore use some of the resampling procedures discussed in the previous +section (as well as one more) in the first stage; we call these resampling procedures uniformπ+imitationX and +restricted-uniformπ+imitationX (the latter of these two uses a permutation scheme that involves g and was +thus not discussed in the previous two sections). These resampling procedures can all be applied to a station- +ary MDP (Environment 4), by using the analogous MDP permutation distribution as described in Section 4.2. +We note here that we do not consider the combinations imitationπ+imitationX, re-imitationπ+imitationX, +and cond-imitationπ+imitationX because, while conditionally i.i.d., they violate the property discussed in +6For uniformπ permutations, we sequentially sample indices uniformly at random from these restricted subsets. +7Note that, practically speaking, if the dataset D in consideration does not have this additional action XT +1, then the +analyst can add it with ease (because they know the adaptive assignment algorithm A) and can do so without affecting the null +or alternative distribution from which the data was drawn (because the null/alternative distributions govern only the transition +dynamics, and not action selection). +17 + +Remark 9 that q(·| ˜D(i)) is the same for all i ∈ [0 : m], and hence require Ω(m2) computations render- +ing them somewhat computationally burdensome. The fourth resampling scheme combines the two stages +of permuting timesteps and randomizing Xt into a single procedure and is thus referred to as combinedπ,X; +this procedure applies in the stationary non-reactive environment (Environment 3) and can be modified to +work in a stationary MDP (Environment 4) by using the usual sequential permutation restriction described +in Section 4.2. Finally, just as in the previous two sections, all of these distributions are based on the idea +of trying to draw resamples in a way that mimics the behavior of A. +imitationX sampling +The imitationX distribution, at each timestep t, conditions on the t − 1 already- +resampled data points ((C1, ˜X1, Y1), . . . , (Ct−1, ˜Xt−1, Yt−1)) as well as Ct and, treating them as though +they were true data points sampled by A, samples ˜Xt amongst all x ∈ X with g(x) = g(Xt), weighting +proportionally to the action-selection probabilities of PA—if all weights are 0, then the process is exited and +an attempt at a new resample can be begun. The intuition behind this sampling procedure is that it attempts +to mimic A by essentially feeding in the already-realized context, g-evaluation, and response sequences +to generate the sequence of actions (each sampled conditionally on the g-evaluation at the corresponding +timestep), thereby mimicking the true data-generating distribution induced by A (resulting in weights closer +to (m + 1)−1). We note here that the imitationX resampling algorithm when applied in Environments 1 and +3 yields precisely the same (unweighted) test as the prior work of Pocock and Simon (1975); Simon (1979); +Bojinov and Shephard (2019); Ham and Qiu (2022). See Algorithm 11 for pseudocode. +uniformπ+imitationX sampling +This resampling procedure proceeds in two stages: the first stage ap- +plies a uniform permutation using the uniformπ sampling of Section 4.1 and the second randomizes the +treatment conditional on its g-evaluation in the permuted data sequence using the imitationX resampling +procedure. Similar to imitationX, we intuitively expect such a sampling procedure to have weights near +(m + 1)−1 while also incorporating significant diversity (resulting in more varied evaluated test statistics +S( ˜D(i))) due to the initial uniform permutation. See Algorithm 12 for pseudocode. +restricted-uniformπ+imitationX sampling +This procedure is identical to uniformπ+imitationX sam- +pling, except that it modifies the uniform sampling in step 1. In particular, for this resampling procedure, +the randomly sampled permutation in the first stage is a restricted uniform permutation, which does not +permute the sequence of g-evaluations. In other words, only permutations π for which g(Xt) = g(Xπ(t)) +for all t ∈ [T] are allowed, and the sampling is uniform over this restricted set. The intuition behind this +sampling scheme is similar to that of the uniformπ+imitationX sampling procedure except that, by using +restricted uniform permutations rather than fully uniform permutations, the sampled data appears more +similar to the original data with the goal of making the weights closer to (m + 1)−1. See Algorithm 13 for +pseudocode. +combinedπ,X sampling +As opposed to the previous two sampling schemes which permute timesteps and +randomize treatments conditional on their g-evaluations in two separate stages, this sampling approach +combines both types of randomization into a single resampling stage. Specifically, at each timestep t, the +combinedπ,X resampling procedure samples t′ according to the imitationπ distribution from Section 4.1 on g- +evaluations over not-yet-selected timesteps, again, by conditioning on the already-resampled data, and then +randomizes Xt′ conditional on g(Xt′). In other words, at timestep t, the combinedπ,X resampling proceeds +by: +1. Selecting a not-already-selected timestep t′ conditionally on ˜Ht−1 via the imitationπ distribution +of Section 4.1 applied to g-evaluations of actions (instead of simply the actions themselves). +If +PA(g(Xs)| ˜Ht−1, Cs) = 08 for all not-yet-selected timesteps s, then no such sample t′ can be gener- +ated and so the sampling process is terminated and a new one may be begun. +2. Once the timestep (Ct′, Xt′, Yt′) is selected, ˜Xt′ is sampled via PA(·| ˜Ht−1, Ct′, g(Xt′)). +8By PA(g(Xs)| ˜Ht−1, Cs) we mean the probability (induced by PA) that the g-evaluation at the tth timestep is equal to +g(Xs) given the history ˜Ht−1 and context Cs. +18 + +Similar to restricted-uniformπ+imitationX sampling, the intuition behind combinedπ,X sampling is that both +randomization across timesteps and the treatments, conditional on their g-evaluations, are incorporated, +except that here they are combined and both the timestep and first action component randomization mimic +A. See Algorithm 14 for pseudocode. +5 +Empirical Results +In Section 4 we proposed a number of resampling procedures q. In this section, we implement these resampling +procedures in a variety of adaptive data collection environments—which are all special cases of the more +general environments discussed in Section 3 for which our theory holds—and both demonstrate their validity +and evaluate their statistical efficiency. +In our experiments, we first study the performance of our tests as the horizon T increases. Thus, we plot +both average Type-I error (to empirically validate the Type-I error control guarantee of Theorem 2.1) as +well as power under an alternative distribution, for values of T ranging from 10 to 100. This power plot, +however, does not paint the whole picture. In each simulation, a combination of two factors influences the +power of our method: 1. the intrinsic difficulty of the environment and 2. the quality of our resampling +algorithm. As an attempt to disentangle these two components, we additionally measure and plot the power +of a standard randomization test on data collected via a baseline uniform i.i.d. adaptive assignment algorithm +which selects actions uniformly at random from X at each timestep t independently from Ht−1. This baseline +measurement captures the intrinsic difficulty of the environment: a less powerful test on data gathered by +this baseline i.i.d. treatment assignment indicates a more difficult environment. Now, while the power plots +do illustrate the second factor regarding the quality of our resampling algorithms, this factor can be further +decomposed into two more contributing components to paint a clearer picture: 1. the effective sample size +of the resampling procedure (i.e., ideally having weights close to (m + 1)−1) and 2. the diversity of the +resamples (i.e., if all resamples are equal—and equal to the original data—then all weights will be (m+1)−1, +but our test will be powerless). We disentangle these two components—and thereby give a more complete +explanation of the accompanying power plots—by measuring effective sample size, +meff := +��m +i=0 wq +D( ˜D(i)) +�2 +�m +i=0 wq +D( ˜D(i))2 , +and plotting meff +m , the fractional effective sample size, under the alternative at the same increments as in +the power plot. In Type-I error, power, and fractional effective sample size plots, we take m, the number +of resamples drawn by our test, to be 100. We also present plots showing how the power of our procedure +grows with m (taking values 102, 103, and 104) for fixed T = 100. +We note here that all randomization tests performed in our simulations use smoothed p-values (see Remark 2) +and thus provably control Type-I error (whose nominal rate is set at α = 0.05 in all our experiments) at +exactly the nominal rate. In addition to testing, we also construct approximate confidence and conformal +prediction intervals (at nominal miscoverage rate 0.05), using the procedures described in Section 3.3; as +these inversion procedures involve non-smoothed p-values, we expect the coverage to be above 0.95. All +results are averaged over 1000 independent trials and plotted with ±2 standard error bars. Finally, we only +show plots involving the weighted MC version of our test (and its inversion for interval construction) in this +section, because for each resampling procedure we consider, our weighted MC test is never outperformed +by (in terms of power and length)—and often in fact dominates—the unweighted MCMC test in all of our +simulations. For analogous plots illustrating results for the unweighted MCMC test, see Appendix G.1. +Computation +We also plot the computation times for each of our resampling algorithms in the various +environments and adaptive data assignment algorithms considered here in Appendix G.2. +In nearly all +environments and assignment algorithms, we are able to run a powerful test on datasets of length T = 100 +within a matter of minutes (and often just seconds) in terms of serial computation time, and our test is +of course embarrassingly parallelizable if desired. +Generally, the computation times for each resampling +19 + +Contextual stationary non-reactive environment +ϵ-greedy, LinUCB +Contextless stationary non-reactive environment +ϵ-greedy, UCB +Contextless C-stationary strongly non-reactive environment +ϵ-greedy, UCB +Contextual C-stationary strongly non-reactive environment +ϵ-greedy, LinUCB +Markov decision process +ϵ-greedy Q-learning, greedy Q-learning +Table 1: Table of environments and adaptive assignment algorithms considered in each. +procedure scale roughly linearly with T. +All code for our simulations is publicly available at https:// +github.com/Yashnair123/RTs-for-AdaptiveData. +Environments and adaptive assignment algorithms +We briefly describe the environments in and +adaptive assignment algorithms for which we conduct our simulations. We consider a total of five different +environments: two are examples of a stationary non-reactive environment (Environment 3)—one with con- +texts and one without contexts—two are instantiations of a C-stationary strongly non-reactive environment +(Environment 5)—again, both contextual and contextless—and the last is an instance of an MDP (Envi- +ronment 2); recall that Environments 3 and 5 are special cases of Environment 1. The adaptive assignment +algorithms we consider in these environments are summarized in Table 1; note that, in each environment, +we consider both a randomized and a deterministic adaptive assignment algorithm. +We now briefly describe each of these adaptive assignment algorithms. Q-learning (Watkins, 1989) maintains +an estimate of the state-action value Q function and selects actions at each timestep based on the current +estimate; we consider one version in which this action selection is (deterministically) greedy and one in which +it is ϵ-greedy. The ϵ-greedy algorithm in the contextless stationary non-reactive environment and contextless +C-stationary strongly non-reactive environment both, at each timestep, determine the empirically best action +and then select an action ϵ-greedily. In the contextual stationary non-reactive environment, the ϵ-greedy +algorithm behaves similarly by maintaining a linear regressor Lx for each action x ∈ X, and progressively +updating Lx with the context-response (i.e., input-output) pair (Ct, Yt) when x is selected at time t (upon +seeing Ct); the algorithm selects, at time t after seeing Ct, the action x with highest predicted response by +the Lx’s. Finally, UCB (Auer et al., 2002) is a deterministic bandit algorithm which additively inflates each +action’s empirical value by a bound on its error with respect to the true value and selects the highest value +action. The contextual analogue in which the response depends linearly on the context is LinUCB (Li et al., +2010). +We note here that essentially all of the adaptive assignment algorithms above are typically used in rein- +forcement learning as they all enjoy quite low regret. However, in comparison to much of the literature +on asymptotic inference in reinforcement learning, which impose clipping constraints on these assignment +algorithms that stipulate that action-selection probabilities cannot be too close to 0 or 1 (Deshpande et al., +2018; Zhang et al., 2020; Hadad et al., 2021), our procedure makes no such assumptions and even allows for +deterministic adaptive assignment algorithms. +5.1 +Conditional independence testing +In this section we apply the conditional independence testing framework and corresponding resampling +algorithms from Section 4.3 to two environments: a stationary strongly non-reactive environment (Environ- +ment 3) with contexts and one without contexts. Our simulations in the former environment demonstrate +the power gain our framework has over prior work (Pocock and Simon, 1975; Simon, 1979; Bojinov and Shep- +hard, 2019; Ham and Qiu, 2022) in a stationary environment by incorporating the additional randomization +over permutations described in Section 3.1. On the other hand, our simulations in the latter environment +focus on the problem of testing if a subset of treatments induce the same response and thus demonstrates +our test’s power on an inferential test for which, to the best of our knowledge, no prior exact test exists. +20 + +Figure 1: Type-I error (left) and power (right) of randomization tests at fixed m = 100 and varying T in a contextual +stationary strongly non-reactive environment on data gathered via ϵ-greedy and LinUCB. +5.1.1 +Conditional independence testing with constant g +Our first series of simulations is in a contextual stationary strongly non-reactive environment, and involves +testing against the null hypothesis H⊥⊥,g with g(Xt) := ∅ (i.e., Yt ⊥⊥ Xt | Ct for all t ∈ [T]). While, as +mentioned in Remark 3, this setting has been covered in prior work, our framework offers a more powerful +test, as our simulation results illustrate. Additionally, we note that in this setting, the weighted MC and +unweighted MCMC tests are identical. +This is because all weights in the MC test are (m + 1)−1 and +all acceptance ratios in the MCMC test are 1 since our resampling procedures all involve the imitationX +distribution which, under a constant g, is the same as sampling, conditionally on the context and response +sequences, from A. For this reason, we do not plot the fractional effective sample sizes, as they too are all +equal to 1. +The precise environment in which our simulations in this section are conducted has treatment space X = +{0, 1}. Letting Ik denote the k × k identity matrix and ⃗1k the length-k vector of all 1’s, we consider null and +alternative distributions involving 2-dimensional contexts Ct sampled i.i.d. from N +�� 1 +−1 +� +, I2 +� +and whose +conditional response distributions are, respectively, given by: +Yt | (Ct, Xt) ∼ N(C⊤ +t ⃗12, 1) and +Yt | (Ct, Xt) ∼ N(C⊤ +t ⃗12 + Xt, 1). +Finally, the test statistic used is simply the absolute value of the t-test statistic against β2 = 0 in a Normal +linear model with design matrix comprising rows of the form (1, Xt, Ct,1, Ct,2) and response vector (Yt)T +t=1. +Figure 1 demonstrates that the added diversity of incorporating uniform permutations—as opposed to sam- +pling only from the imitationX distribution—indeed results in an increase in power. Notably, when comparing +our resampling scheme, uniformπ+imitationX, against the imitationX resampling of prior work (Pocock and +Simon, 1975; Simon, 1979; Bojinov and Shephard, 2019; Ham and Qiu, 2022) we observe that our approach +is more powerful than theirs on data collected via an ϵ-greedy treatment assignment. Perhaps even more +strikingly, our approach when applied to LinUCB, a deterministic adaptive assignment algorithm, has high +and increasing (with T) power. This is in contrast to the usual imitationX resampling of the prior work, +which is powerless. We relegate the power curves for increasing m but finite T to Appendix G.3 as they are +all approximately constant. +21 + +Contextual stationary strongly non-reactive environment conditional independence test +0.14 +1.0 +0.12 +0.8 +点 0.10 +rpe-l +0.08 +a6 +0.04 +0.2 +0.02 +0.00 +0.0 +20 +40 +Q9 +08 +100 +20 +40 +60 +80 +T +T +uniformiidbaseline +LinUCB uniformn+imitationx +-greedy imitationx (priorwork) +LinUcB imitationx(priorwork) +-greedyuniformn+imitationxFigure 2: Type-I error rate (leftmost) and power (second from left) of the MC randomization test at fixed m = 100 +and varying T as well as power at fixed T = 100 and varying m (third from left) and fractional effective sample size +plots at fixed m = 100 and varying T (rightmost) in a contextless stationary strongly non-reactive environment on +data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +5.1.2 +Conditional independence testing with non-constant g +We now discuss our simulations in the contextless stationary non-reactive setting of Environment 3, given +as an example in the beginning of Section 3.1. +In particular, we consider a contextless instantiation of +a stationary non-reactive environment with action space X = {0, 1, 2} and are testing against the null +hypothesis H⊥⊥,g +0 +with g(Xt) := I(Xt = 2) (i.e., that Yt ⊥⊥ Xt | Xt ∈ {0, 1} for all t ∈ [T]). +In this environment, we specify the null and alternative distributions as +Yt | Xt ∼ +� +N(0, 1) if Xt ∈ {0, 1} +N(2, 1) if Xt = 2 +and +Yt | Xt ∼ +� +� +� +� +� +N(0, 1) if Xt = 0 +N(3, 1) if Xt = 1 +N(2, 1) if Xt = 2 +, +respectively. +In all our simulations, the test statistic S(D) we use is, similar to as in Section 5.1.1, simply the absolute +value of the t-test statistic for the test against β2 = 0 in a Normal linear model whose design matrix has +(1, I(Xt = 0), I(Xt = 2)) as its tth row and (Yt)T +t=1 as the response vector. Figure 2 summarizes the results +in this domain. +First, the Type-I error rates in Figure 2 are controlled at exactly the nominal level, validating the theoretical +guarantee of Theorem 2.1. With regards to power, we notice that restricted-uniformπ +imitationX sampling +performs better on data gathered via ϵ-greedy while combinedπ,X sampling performs better under UCB. +The fractional effective sample size plots in the same figure offer an explanation for why this is the case: the +fractional effective sample sizes under ϵ-greedy are larger with restricted-uniformπ+imitationX sampling than +with combinedπ,X and, conversely, under UCB they are larger with combinedπ,X than restricted-uniformπ + +imitationX. We note here that this testing problem is much harder on data gathered via ϵ-greedy and UCB +than on the uniform i.i.d. baseline (thus explaining the power gap between the latter and all resampling +procedures applied on data gathered by the former two, especially for large T). This is because ϵ-greedy +and UCB are low-regret adaptive assignment algorithms and thus, under the alternative, will select action +1 very frequently, and all other actions much less frequently. Hence the problem of detecting if the response +distributions induced by Xt = 0 and Xt = 1 are the same becomes much more challenging, as action 0 is +sampled rarely under these two low-regret algorithms. On the other hand, it is sampled more frequently and +at the same rate as action 1 under the uniform i.i.d. baseline. Despite this challenge, however, our method +is still able to attain quite high power under both ϵ-greedy and UCB adaptive assignment algorithms. +22 + +Contextless stationary strongly non-reactive environment conditional independence test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 +0.125 +0.8 +0.8 +(Altemative) +0.8 +Power +Power +0.100 +0.6 +0.6 +0.6 +0.075 +Average +0.4 +0.4 +0.4 +0.050 +mm +0.2 +0.2 +0.2 +0.000 +0.0 +25 +0.0 +0.0 +25 +50 +75 +100 +50 +75 +102 +103 +104 +25 +50 +75 +100 +T +T +m +T +uniform idbaseline +e-greedy uniformn+imitationx +UCB restricted-uniformn+imitationx +E-greedyrestricted-uniformn+imitationx +UCB combinedn,x +UCB uniformn+imitationx +-greedy combinedn,xFigure 3: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 +and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size +at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with +data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +5.2 +Non-stationarity testing +In this section, we empirically evaluate our test of non-stationarity on three different environments: the first +two are examples of the C-stationary strongly non-reactive setting of Environment 5—one with contexts and +one without—and the third is an MDP (Environment 2). +5.2.1 +Testing non-stationarity in a C-stationary strongly non-reactive environment +Our simulations in this section are performed on a contextless C-stationary strongly non-reactive environment +with action space X = {−1, 1} and Gaussian rewards: the reward distribution for the first T − 1 steps is +given by Yt | Xt ∼ N(Xt, 1). Under the null hypothesis HS +0, the reward distribution at the T th timestep +is unchanged and hence YT | XT ∼ N(XT , 1). We analyze power under an alternative distribution that +samples YT | XT ∼ N(4XT , 1). Finally, as we are testing for non-stationarity, our test statistic S is simply +a non-conformity score, and we choose it to be the absolute residual: S(D) = |YT − ˆµD(XT )|, where ˆµD is +the fitted ordinary least squares (OLS) model to D with an intercept term. +Figure 3 shows the results for our simulations in this section. Indeed the Type-I error plots once again +demonstrate the validity our randomization tests. In terms of power, Figure 3 shows that cond-imitationπ +sampling performs best under an ϵ-greedy treatment assignment and imitationπ performs best under UCB +and both attain power close to that of the uniform i.i.d. baseline for nearly all values of T. +Figure 3, +however, also shows that, with large enough m, re-imitationπ sampling eventually performs better than +cond-imitationπ at T = 100. Combining this information with the fractional effective sample size plots of +Figure 3, we thus see that while cond-imitationπ has greater effective sample size than re-imitationπ, it +has less diverse samples, leading to a more powerful procedure when m is small (since, in this regime, the +diversity plays a greater role), but a (slightly) less powerful one when m is large. +5.2.2 +Testing non-stationarity in a contextual C-stationary strongly non-reactive environment +We now describe our simulations in a contextual C-stationary strongly non-reactive environment. +The +specific environment we consider in this section involves a action space X = {−1, 1} and 100-dimensional +contexts sampled i.i.d. from N(⃗1100, I100). +The conditional response distribution during the first T − 1 +23 + +Contextless c-stationary strontly non-reactive environment, non-stationarity test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 - +1.0 +1.0 +0.8 +Power +0.8 +0.8 +Power +0.100 +Type- +0.6 +0.6 +0.6 +0.075 +Average +ae +0.4 +0.4 +0.4 +0.050 +mm + 0.025 +0.2 +0.2 +0.2 +0.000 +75 +0.0 +0.0 +0.0 +25 +50 +100 +25 +50 +75 +102 +103 +104 +25 +50 +75 +T +T +m +T ++- uniform iid baseline +-greedyimitationn +E-greedy cond-imitationn +UCBimitationn +-greedyuniformr +-greedy re-imitationn +.....UCBuniformnFigure 4: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 +and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size +at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with +data gathered via ϵ-greedy, LinUCB, and the uniform i.i.d. baseline. +timesteps is a sparse linear combination of the context vector and is given by +Yt | (Ct, Xt) ∼ N(−5Xt + +10 +� +j=1 +Ct,j, 1). +The null conditional response distribution at time T is of course the same as the above whereas the alternative +distribution we evaluate power under swaps the effects of the two treatments and is given by +YT | (CT , XT ) ∼ N(5XT + +10 +� +j=1 +CT,j, 1). +We regularize the regressor Lx used in the ϵ-greedy adaptive assignment by using Lasso regression with with +penalty parameter 10 as opposed to OLS. Finally, the test statistic used is the same non-conformity score as +in the previous section, except that we again use Lasso—this time with penalty parameter chosen through +5-fold cross-validation—instead of OLS, as ˆµD. +Figure 4 shows the Type-I error, power, and effective sample size curves for our simulations in this envi- +ronment. The Type-I error is again controlled at the nominal level. In terms of power, similar conclusions +to those drawn in the contextless C-stationary strongly non-reactive environment of the previous section +apply here, too. In particular, while cond-imitationπ sampling performs best under an ϵ-greedy adaptive +assignment at m = 100 (and again exhibits, for large T, power quite close to—and, in fact greater than, +for small T—that of the uniform i.i.d. baseline) it has essentially the same power as re-imitationπ at both +m = 103 and m = 104. Using the fractional effective sample size plots, we may therefore infer, once again, +that while cond-imitationπ sampling has a higher effective sample size than re-imitationπ does under ϵ- +greedy, its samples exhibit less diversity than those of re-imitationπ. Under LinUCB, the contextual analog +of the deterministic UCB assignment, however, we see that while imitationπ has quite high power for small +to moderate values of T, the power drops for larger T. The fractional effective sample size curve explains +that this is caused by a corresponding drop in the effective sample size as T grows. We hypothesize that +this is due to the extremely uneven action selection exhibited by LinUCB due to its very low regret (even +as compared to ϵ-greedy); we discuss why this low regret and the uneven action selection it causes—which +we again emphasize is extreme in the case of LinUCB, even in comparison to ϵ-greedy—renders the testing +problem harder in the paragraph below. As such, we leave the problem of developing a powerful resampling +scheme for deterministic algorithms like LinUCB in this type of contextual C-stationary strongly non-reactive +environment to future work. +24 + +Contextual c-stationary strontly non-reactive environment, non-stationarity test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 +0.8 +0.8 +0.8 +Power +Power +0.100 +Type- +0.6 +0.6 +0.6 +0.075 +ae +abe +0.4 +0.4 +0.4 +0.050 +mm + 0.025 +0.2 +0.2 +0.2 +0.000 +75 +0.0 +0.0 +0.0 +25 +50 +100 +25 +@5 +75 +102 +103 +104 +25 +50 +75 +m +T ++-uniformidbaseline +-greedy imitationr +-greedy cond-imitationm +...LinUCBimitationn +-greedy uniformr +-greedy re-imitationn +.....LinUCB uniformnLastly, we hypothesize that the shift in relative power of the uniform i.i.d. baseline (from the least powerful +adaptive assignment-resampling algorithm combination for small T to the most powerful for large T) is again +an artifact of the low-regret properties of ϵ-greedy and LinUCB. This low regret results in the outlier context- +action-response triple sampled at the T th timestep to often have action agreeing with the action taken during +most of the first T − 1 timesteps. Thus, for small T, these first T − 1 context-treatment-response triples +may outweigh the effect of the outlier at time T when training ˆµD via Lasso with cross-validation, thereby +resulting in better outlier detection than that under the uniform i.i.d. base, wherein only around half of the +first T − 1 timesteps (and thus very few, in total) will have action agreeing with that at the T th. The effect +however, is diminished with large T, most likely because in that regime even half of the data generated via +Yt’s true (null) conditional distribution, which agrees with the action taken at time T, is enough to offset +the effect of the outlier in training ˆµD. In particular, the remaining timesteps whose corresponding action +differs from the one taken at time T—of which there are more under the uniform i.i.d. baseline than under +ϵ-greedy (and many more under the baseline than under LinUCB) for large T—may allow for a better fit ˆµD +through more effective learning of the dependence of Yt on Xt. As noted above, this may also explain the +difficulty in attaining high power for large T under LinUCB in this environment. We empirically validate +this hypothesis in Figure 24 of Appendix G.3 by showing that the uniform i.i.d. baseline exhibits precisley +this same relative performance, with respect to T, in comparison to a biased i.i.d. assignment algorithm that +selects action −1 with probability 0.9 and otherwise selects action 1. +5.2.3 +Testing non-stationarity in an MDP +Our final set of hypothesis testing simulations is in an MDP, where recall that Yt = Ct+1 and hence we +drop the notation Yt and refer to Ct as states instead of contexts.9 The precise specifications of environment +involve a state space of C = {0, 1, 2}, action space X = {−1, 1}, and transition kernel during the first T − 1 +transitions given by +Ct+1 | (Ct, Xt) ∼ +� +Ct + Xt +(mod 3) with probability 0.95 +Ct − Xt +(mod 3) with probability 0.05 +. +Under the null hypothesis, the transition distribution at time T remains the same as above, but under the +alternative, we swap the role of the two actions so that the alternative distribution is given by +CT +1 | (CT , XT ) ∼ +� +CT − XT +(mod 3) with probability 0.95 +CT + XT +(mod 3) with probability 0.05 +. +Finally, the test statistic that we use in these simulations trains a decision tree classifier on the data D and +outputs the negative log likelihood loss of the trained model on the triple (CT , XT , CT +1) at time T. +Figure 5 summarizes the results for our simulations in this setting. Once again, Type-I error is controlled +at exactly the nominal level, as guaranteed by our theory. In a departure from the results of the previous +two sections (most likely due to the sequential dependence in this environment not present in the last two +as well as the modified sampling process that it requires), uniformπ resampling has the greatest power for +Q-learning with ϵ-greedy action selection. Additionally, both uniformπ and imitationπ resampling perform +similarly well for Q-learning with greedy action selection. Again, using both the fractional effective sample +size plots and power plots with fixed T but varying m, we see that under ϵ-greedy, re-imitationπ resampling +has lower effective sample size than uniformπ, but higher diversity, leading it to perform worse for smaller +m but better for large m. Finally, we note that, in comparison to the uniform i.i.d. baseline10, uniformπ +9As discussed in Section 4.2, the dataset we consider in these simulations is the usual dataset D along with one final action +taken at time T + 1: ((C1, X1, C2), . . . , (CT , XT , CT +1), XT +1) as this allows us to simply permute the state-action pairs +(Ct, Xt). +10In contrast to the last section, the uniform i.i.d. data used in this section is not actually gathered in the same environment +as the other adaptive assignment algorithms, and in particular, the uniform i.i.d. data is not MDP data. Rather, the data +comprises state-action-next state triples (C, X, C′) sampled i.i.d. from a distribution in which both C and X are independent +and uniform over C and X respectively, and C′ is sampled from the transition distribution described above, conditional on +(C, X). +25 + +Figure 5: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 +and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size +at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with +data gathered via ϵ-greedy and greedy Q-learning. +and imitationπ are very competitive under the greedy Q-learning adaptive assignment, and imitationπ also +exhibits quite high power under ϵ-greedy. +5.3 +Constructing confidence and prediction intervals +In this section we apply our framework to constructing confidence and prediction intervals using the inversion +procedures described in Section 3.3. In particular, we plot both the coverage of the intervals as well as their +average length, averaged over 1000 trials for varying T at a fixed number of MC samples m = 100. For +construction of conformal intervals, we also apply the sample sharing described in Remark 7 (in addition +to the standard gridding procedure described in Section 3.3) at both m = 10 and m = 100. Finally, we +note that some of the resampling procedures in certain environments and adaptive assignment algorithms +considered in this section exhibit overcoverage, in contrast to the last section, in which all tests attained +Type-I error control at exactly the nominal level. We remind the reader that this is due to the fact that the +gridding procedure described in Section 3.3 is based on the (approximate) inversion of a conservative test, +rather than that of the smoothed test described in Remark 2. +5.3.1 +Confidence intervals +We now discuss our simulations constructing confidence intervals using the gridding procedure described at +the end of Section 3.3. We consider essentially the same environment as the contextless stationary strongly +non-reactive environment discussed in Section 5.1.2, except that here we have +Yt | Xt ∼ +� +� +� +� +� +N(0, 1) if Xt = 0 +N(b0, 1) if Xt = 1 +N(2, 1) if Xt = 2 +, +with b0 = 4; we construct a confidence interval for the location difference between Yt | (Xt = 0), and +Yt | (Xt = 1), which simply corresponds to b0, using a gridding set Y′ = [−1 : 9]. +Figure 6 summarizes the results for these simulations. We note there is no evidence in our simulations that the +necessary (and in fact rather coarse) gridding of Y results in any undercoverage. In terms of length, our results +align with the power results presented in Section 5.1.2. In particular, under an ϵ-greedy adaptive assignment, +26 + +MDP, non-stationarity test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 . +Power +0.8 +Power +0.8 +0.100 +Type-l +0.6 +0.6 +0.6 +0.075 +Average +Average +(Alter +0.4 +0.4 +0.4 +0.050 +mim +0.025 +0.2 +0.2 +0.2 +0.000 +0.0 +0.0 +0.0 +25 +50 +75 +100 +25 +50 +75 +100 +102 +103 +104 +25 +50 +75 +100 +T +T +m +T ++-uniformiidbaseline +-greedy cond-imitationr ++ -greedy uniformn +greedy uniformr +-greedyimitationn +greedy imitationn +-greedy re-imitationnFigure 6: Coverage and average length of confidence intervals for location difference between Yt | (Xt = 0), and +Yt | (Xt = 1) (i.e., b0 = 4) +using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +restricted-uniformπ+imitationX sampling produces the shortest average length intervals, whereas under +UCB, combinedπ,X resampling does. +5.3.2 +Conformal prediction intervals +Finally, we discuss our simulations constructing conformal prediction intervals. The environment considered +in this section is the same as contextless C-stationary strongly non-reactive environment considered in +Section 5.2.1 except that we do not have access to YT ; it is the quantity for which we construct the conformal +prediction region. +Our simulations in this section use the gridding procedure described at the end of Section 3.3 and assess the +benefit of the sharing of samples described in Remark 7 and Appendix E.2. Without sample sharing, we fix +m = 100 and set Y′ = [−5 : 5] and plot coverage and length at varying T. With sample sharing, we use the +same gridding set Y′ and study both m = 10 and m = 100 number of MC samples11 and varying T. As +discussed in Appendix E.2 this sample sharing results in non-conditionally-i.i.d. draws because the resamples +generated corresponding to y1 ∈ Y′ may come from (and indeed do in our simulations) a different distribution +than those sampled according to y2 ∈ Y′ for y1 ̸= y2. As such, following Remark 8 (and as discussed in +further detail in Appendix E.1), we choose a subset of permutations Σ to be the set of m + 1 permutations +swapping 0 and i for each i ∈ [0 : m] and employ the test of Algorithm 1 with weights w˜q,Σ +D ( ˜D(i)). +Figures 7 and 8 summarize our simulation results constructing approximate conformal prediction intervals. +Figure 7 illustrates both coverage and length as T grows by simply using the standard gridding procedure +described at the end of Section 3.3. With regards to average length, our results mirror those in Section 5.2.1, +in which cond-imitationπ resampling is best for an ϵ-greedy adaptive assignment, and imitationπ is best +for UCB. In Figure 8, we observe that any undercoverage, attributable to the gridding of Y, is relatively +minor, and should be fixable by using a finer-resolution grid; the reason that the uniform i.i.d. baseline tends +to exhibit the greatest undercoverage is most likely explained by that fact that we are not smoothing in +this set of experiments, so that whereas all other procedures are conservative, the uniform i.i.d. baseline +is not. Additionally, we see essentially the same ranking of resampling procedures, although re-imitationπ +does perform slightly better than cond-imitationπ when 100 MC samples are used. Finally, the shortest +11For each y ∈ Y′, m samples are generated, but all m|Y′| samples are used to determine the membership of y ∈ Y′ in the +prediction interval +27 + +Confidence interval +1.00 +10 +0.98 +8 +96'0 +Coverage +Average Length +0.94 +0.92 +0.90 +0.88 +0.86 +20 +40 +60 +100 +80 +20 +40 +60 +80 +100 +T +T +uniformiidcomparator +-greedy uniformn+imitationx +UCB restricted-uniform,+imitationx +E-greedy restricted-uniformn+imitationx +UCB combinedn,x +UCB uniform,+imitationx +-greedy combinedr,xFigure 7: Coverage and average length of conformal prediction intervals for YT using the MC randomization test +with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +Figure 8: Coverage and average length of approximate conformal prediction intervals, with sample sharing, for YT +using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +28 + +Contextless C-stationary strongly non-reactive environment, conformal prediction interval +1.00 +10 +0.98 +8 +0.96 +Average Length +Cov +0.94 +0.92 +ferage +06'0 +A +0.88 +0.86 +20 +40 +09 +80 +100 +uniformiidbaseline +E-greedyimitationn +E-greedy cond-imitationn +UCB imitationn +E-greedy uniformn +E-greedy re-imitationm +UCB uniformnContextual c-stationary strongly non-reactive environment, conformal prediction interval, shared samples +1.00 +1.00 +QT +10 +(00T =) +Coverage +0.95 +Coverage +0.95 +T +4 +4 +0.90 +0.90 +Average +N +0.85 +0.85 +2550 +25 +50 +75 +100 +[0] +25 +50 +75 +100 +[0] +75 +100 +25 +5075 +100 +T +T +T +T +uniform iid baseline +E-greedy cond-imitationn +E-greedy uniformn +-greedy imitationn +UCB imitationn +UCB uniformr +E-greedy re-imitationnaverage length curves in Figure 8 using m = 10 are quite competitive in comparison to their counterparts in +Figure 7. In particular, imitationπ under UCB produces only slightly wider intervals with m = 10 and sample +sharing than with m = 100 and no sample sharing; for ϵ-greedy, cond-imitationπ performs essentially the +same in both settings. This demonstrates that, by utilizing sample sharing, we are able to obtain conformal +prediction intervals of nearly the same length at a fraction of the computational cost (because in the left +hand column of Figure 8 we generate a total of 110 samples, whereas in Figure 7, we generate 100 samples for +each y ∈ Y′). Indeed, as illustrated by Figures 21 and 22 of Appendix G, the computation time required by +the procedure using m = 10 and shared sampling is, for most resampling schemes and adaptive assignments, +more than 5 times faster than its m = 100 non-sample sharing counterpart. +6 +Discussion +In this paper, we study the problem of performing various challenging inferential tasks on adaptively collected +data. Through the development of a weighted MC randomization test (along with the novel application of +the unweighted MCMC randomization test of Besag and Clifford (1989)), we show that such tasks can be +performed with exact Type-I error control with a great degree of flexibility in the choice of resampling +procedure as long as the proportionality hypothesis H∝ +0 is satisfied. +The question, however, of how best to perform these tests still remains, and is an interesting one for future +research. In particular, while we have discussed some preliminary empirical results demonstrating powerful +resampling algorithms in a number of different challenging environments, there are a number of interesting +and potentially fruitful directions forward. +In particular, as mentioned in Section 5.2.2, one important +direction for future work is to develop resampling procedures that are powerful for deterministic algorithms +like LinUCB in a contextual C-stationary strongly non-reactive setting. +But more generally, it may be +interesting to formally investigate any commonalities between powerful resampling algorithms in different +environments in order to develop a more principled method for deciding which procedure to apply to a +particular problem (i.e., combination of adaptive assignment algorithm, environment, and inferential task). +Along this same vein of how best to resample, it may be useful to also consider different resampling procedures +which incorporate non-conditionally-i.i.d. sampling. Utilization of these non-conditionally-i.i.d. resampling +schemes under the unweighted MCMC randomization testing framework has the potential to be especially +advantageous as the full MCMC test will still require only O(m) number of computations. On the other +hand, as discussed in Remark 8 and and Appendix E.1, when applying such sampling schemes to the weighted +MC test, we may generalize and choose a subset Σ of the full permutation set Π[0:m] different than Σswap, +and condition only on the data permutations induced by these sets rather than the full set of permutations. +As such, one final direction for future work that may also be of interest is to investigate how the power of +the weighted MC test changes under different choices of Σ. Specifically, while we expect the test to be more +powerful for choices of Σ which are larger, we also expect such larger choices to require a greater amount of +comptutation time. 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URL https://arxiv.org/abs/2203. +10980. +33 + +A +Proof of Theorem 2.1 +We consider the case of discrete data D and discrete resamples ˜D(1), . . . , ˜D(m) and apply Bayes’ theorem. +Define d = (d0, . . . , dm) to be be some list of data sets, d−0 to be the list (d1, . . . , dm), D−0 to denote the +list ( ˜D(1), . . . , ˜D(m)), and orb(d) := {dπ : π ∈ Π[0:m]}, where dπ := (dπ(0), . . . , dπ(m)); we also use (d−0)π to +denote the permuted list (dπ(1), . . . , dπ(m)) for any π ∈ Π[0:m]. Then, by Bayes’ theorem, we have that +P (D = di|D ∈ orb(d)) = +P (D ∈ orb(d)|D = di) f(di) +� +distinct dj∈d P (D ∈ orb(d)|D = dj) f(dj) += +P +� +D−0 ∈ {(d−0)π : π ∈ Π[0:m] such that dπ(0) = di}|D = di +� +f(di) +� +distinct dj∈d P +� +D−0 ∈ {(d−0)π : π ∈ Π[0:m] such that dπ(0) = dj}|D = dj +� +f(dj) += +|{(d−0)π : π ∈ Π[0:m] such that dπ(0) = di}| +�� +k∈[0:m]\{i} q(dk|di) +� +f(di) +� +distinct dj∈d |{(d−0)π : π ∈ Π[0:m] such that dπ(0) = dj}| +�� +k∈[0:m]\{j} q(dk|dj) +� +f(dj) +. +where we say that d ∈ d if d is an element of the list d and where the last equality follows from the conditional +i.i.d.-ness of the resamples given D. +Letting mdi(d) denote the number of times di appears in the list d we have that +|{(d−0)π : π ∈ Π[0:m] with dπ(0) = di}| = +m! +(mdi(d) − 1)! +� +distinct d∈d not equal to di +md(d)! += mdi(d) · +m! +� +distinct d∈d +md(d)! +and hence, +P (D = di|D ∈ orb(d)) = +mdi(d)f(di) � +k∈[0:m]\{i} q(dk|di) +�m +j=0 f(dj) � +k∈[0:m]\{j} q(dk|dj) += +mdi(d) ˆf(di) � +k∈[0:m]\{i} q(dk|di) +�m +j=0 ˆf(dj) � +k∈[0:m]\{j} q(dk|dj) += mdi(d)wq +d(di), +where the second-to-last inequality is because of the proportionality hypothesis H∝ +0 . Equivalently, we can +think of D’s conditional distribution given orb(D) as a draw from the m + 1 elements of D, with weight on +each element given by the wq +D function applied to that element. Defining the set S := {S( ˜D) : ˜D ∈ D}12, +S(D) can be viewed as a draw from the weighted distribution on S with the total weight of each element +S( ˜D(i)) equal to +� +j:S( ˜ +D(j))=S( ˜ +D(i)) +ˆf( ˜D(j)) � +k∈[0:m]\{j} q( ˜D(k)| ˜D(j)) +�m +j′=0 ˆf( ˜D(j′)) � +k′∈[0:m]\{j′} q( ˜D(k′)| ˜D(j′)) += +� +j:S( ˜ +D(j))=S( ˜ +D(i)) +wq +D( ˜D(j)). +Finally, note that +P(p ≤ α) = P +� m +� +i=0 +wq +D( ˜D(i))1[S( ˜D(i)) ≥ S(D)] ≤ α +� += P +� +�S(D) > inf{s ∈ S : +� +˜ +D∈D:S( ˜ +D)≤s +wq +D( ˜D) ≥ 1 − α} +� +� . +12Just as above, we say that ˜D ∈ D if ˜D is an element of the list D. +34 + +Then, we have that +P +� +�S(D) > inf{s ∈ S : +� +˜ +D∈D:S( ˜ +D)≤s +wq +D( ˜D) ≥ 1 − α} +������ +orb(D) +� +� ≤ α, +(11) +since 1. the infimum in equation (11) is the (1 − α)th conditional quantile of the weighted distribution over +S and 2. as noted above, S(D) is conditionally a draw from this weighted distribution. Inequality (11) then +also holds unconditionally, as desired. +B +Conditional independence testing in a stationary MDP +In this section, we give a proof of Proposition 3.2 regarding the restricted permutation set in Environment 4: +Proof. Let πi be any permutation in +Π4 := {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1}. +Then +ˆf( ˜D(i)) = +T +� +t=1 +PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) += +�T +t=1 P(Yπi(t)|Cπi(t), Xπi(t))PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) +�T +t=1 P(Yt|Ct, Xt) +because of equation (6) += P(Cπi(1)) �T +t=1 P(Cπi(t+1)|Cπi(t), Xπi(t))PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) +P(C1) �T +t=1 P(Ct+1|Ct, Xt) +since πi ∈ Π4 += P(Cπi(1)) �T +t=1 P(Cπi(t+1)|Cπi(t), ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) +P(C1) �T +t=1 P(Ct+1|Ct, Xt) +by H⊥⊥,g +0 +and that g( ˜X(i) +t ) = g(Xπi(t)) += P( ˜C(i) +1 ) �T +t=1 P( ˜C(i) +t | ˜C(i) +t , ˜X(i) +t )PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) +P(C1) �T +t=1 P(Ct+1|Ct, Xt) +by equation (8) += +1 +P(C1) �T +t=1 P(Ct+1|Ct, Xt) +· f( ˜D(i)), +due to equation (1) +and so the proportionality hypothesis is satisfied with K = +1 +P(C1) �T +t=1 P(Ct+1|Ct,Xt). +C +Inference in a general adaptive data collection process +In this section, we relax all structural assumptions made in Section 3 and assume that the data are gathered +according to any adaptive data collection process (i.e., we make no assumptions on the joint distribution +of (Ct, Xt, Yt)T +t=1 other than that the sequence (Xt)T +t=1 is non-anticipating with respect to the filtration +Ft := σ (Ht−1 ∪ {Ct})). The hypothesis we are interested in testing is related to that discussed in Section 3.1, +except here it involves sequences of actions. In fact, we consider T functions g1, . . . , gT , where gt takes as +input the sequence of first t actions X1:t := (X1, . . . , Xt); the null hypothesis we are interested in testing is +that of simultaneous independence, conditional on these g-evaluations: +H⊥⊥,g1,...,gT +0 +: [Ct ⊥⊥ X1:t−1 | (C1:t−1, gt−1(X1:t−1), Y1:t−1)] and [Yt ⊥⊥ X1:t | (C1:t, gt(X1:t), Y1:t−1)] , ∀t ∈ [T], +where we define +C1:0 = X1:0 = g0(X1:0) = Y1:0 = ∅. +35 + +Informally, H⊥⊥,g1,...,gT +0 +states that the actions Xt only influence future contexts and responses via their +values filtered through the functions gt. As noted in Remark 4, the setting in which all gt are constant has +been covered in prior work by Bojinov and Shephard (2019); however, to the best of our knowledge, the case +of non-constant gt is novel. We now describe and prove the validity of the testing procedure for the null +hypothesis H⊥⊥,g1,...,gT +0 +. +Proposition C.1. Suppose that the resampling distribution q randomizes only the action sequence X1:T +conditional on (gt(X1:t))T +t=1. That is, in each resample ˜D(i), we have that +� +˜C(i) +t , gt( ˜X(i) +1:t), ˜Y (i) +t +� += (Ct, gt(X1:t), Yt) , ∀t ∈ [T]. +(12) +Then the null hypothesis H⊥⊥,g1,...,gT +0 +implies the proportionality hypothesis H∝ +0 . +Proof. We have that +ˆf( ˜D(i)) += +T +� +t=1 +PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1) += +�T +t=1 P( ˜Y (i) +t +| ˜C(i) +1:t, gt( ˜X(i) +1:t), ˜Y (i) +1:t−1)PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P( ˜C(i) +t | ˜C(i) +1:t−1, gt−1( ˜X(i) +1:t−1), ˜Y (i) +1:t−1) +�T +t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) +equation (12) += +�T +t=1 P( ˜Y (i) +t +| ˜C(i) +1:t, gt( ˜X(i) +1:t), ˜Y (i) +1:t−1, ˜X(i) +1:t)PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P( ˜C(i) +t | ˜C(i) +1:t−1, gt−1( ˜X(i) +1:t−1), ˜Y (i) +1:t−1, ˜X(i) +1:t−1) +�T +t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) +due to H⊥⊥,g1,...,gT +0 += +�T +t=1 P( ˜Y (i) +t +| ˜C(i) +1:t, ˜X(i) +1:t, ˜Y (i) +1:t−1)PA( ˜X(i) +t | ˜C(i) +t , ˜H(i) +t−1)P( ˜C(i) +t | ˜C(i) +1:t−1, ˜X(i) +1:t−1, Y (i) +1:t−1) +�T +t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) += +1 +�T +t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) +· f( ˜D(i)), +which satisfies the proportionality hypothesis with K = +1 +�T +t=1 P(Yt|C1:t,gt(X1:t),Y1:t−1)P(Ct|C1:t−1,gt−1(X1:t−1),Y1:t−1). +One concrete setting in which the above randomization procedure could be used to test against the hypothesis +H⊥⊥,g1,...,gT +0 +is in a completely general mobile health setting in which no environmental assumptions are made +and there are only two treatments: a control and an experimental treatment. A hypothesis which may be of +interest to test is that, conditional on the prior sequence of contexts and responses, each of the next response +and context is independent of the action sequence given the number of experimental treatments taken up +until that time. In such a setting, one could use our framework under the above randomization scheme to test +the null hypothesis H⊥⊥,g1,...,gT +0 +where gt(X1:t) is equal to the number of times the experimental treatment +appears in the treatment sequence X1:t; i.e., gt(X1:t) = �t +s=1 Xs where X = 1 denotes the experimental +treatment and X = 0 denotes the control. +D +Inference in scale families +In this section, we describe how the test discussed in Section 3.3.1 can also be used for inference in semi- +parametric scale families. +Instead of assuming that each reward distribution is a member of the same location family, it may be more +natural in some situations to assume a semiparametric scale family. As such, letting θx now denote action +x’s scale parameter (i.e., we assume that Yt | Xt = x ∼ h0(y/θx)13), we may instead test against the null +13Recall that we only consider discrete random variables, per Remark 1, and hence no Jacobian is required. +36 + +hypothesis HScale,δ,x,x′ that θx′ +θx = δ. To do so, we simply revise the reward-modification trick discussed in +Section 3.3.1 for the location family to instead replace Yt with Yt · δ · 1[Xt = x] and again perform the usual +test for conditional independence from Section 3.1 using g(Xt) = +� +{x, x′} if Xt ∈ {x, x′} +Xt otherwise +. +Similar to as discussed in Section 3.3.1, the above test can be inverted to construct (simultaneous) confidence +intervals. +E +Non-conditionally-i.i.d. resampling and sharing samples +In this section, we discuss how to handle non-conditionally-i.i.d. resampling procedures and how this more +general procedure can be used to share samples between different values y ∈ Y in the construction of +conformal prediction regions. +E.1 +Non-conditionally-i.i.d. resampling +In this section, we describe a more general version of the weighted MC randomization test of Algorithm 1 +that can be used in the setting of non-conditionally-i.i.d. resampling, as discussed in Remark 8. In particular, +the Remark states that, letting ˜q denote the joint conditional distribution of the resamples ( ˜D(1), . . . , ˜D(m)), +one may redefine the weights to be +w˜q,Σ +D ( ˜D(i)) = +ˆf( ˜D(i)) � +π∈Σ:π(0)=i ˜q((D−0)π| ˜D(i)) +� +j∈[0:m] ˆf( ˜D(j)) � +π′∈Σ:π′(0)=j ˜q((D−0)π′| ˜D(j)) +, +(13) +and the p-value +p := +m +� +i=0 +w˜q,Σ +D ( ˜D(i))1[S( ˜D(i)) ≥ S(D)] +will be valid. +Recall that the key idea behind Algorithm 1—and the proof of Theorem 2.1—was to condition on the event +( ˜D(0), . . . , ˜D(m)) ∈ {(dπ(0), . . . , dπ(m)) : π ∈ Π[0:m]} for some list d = (d0, . . . , dm) and to apply Bayes’ +theorem as well as the proportionality hypothesis to derive the weight formula (equation (3)). The entire +permutation set Π[0:m] need not, however, be considered. Our generalization involves using any arbitrary +subset of permutations Σ; when Σ is small, our method is computationally tractable for non-conditionally- +i.i.d. resampling schemes. +More specifically, define +pseudo-orbΣ(d) := {(dπ(0), . . . , dπ(m)) : π ∈ Σ} +to be the Σ-pseudo-orbit14 of the list d. Then, by conditioning on pseudo-orbΣ(D), we will be able to show +that the weighting given by equation (13) will be valid. We first prove a result similar to Theorem 2.1 in the +case of distinct data D and resamples ˜D(1), . . . , ˜D(m) using exactly the same proof strategy: +Lemma E.1. Let Σ be any subset of the full set of permutations on [0 : m], Π[0:m]. +Define Σi to be +{π ∈ Σ : π(0) = i} and assume that ˜q is such that ˜D(0), . . . , ˜D(m) are all distinct. Then, with weights +w˜q,Σ +D,i( ˜D(i)) := +ˆf( ˜D(i)) � +π∈Σi ˜q((D−0)π| ˜D(i)) +�m +j=0 ˆf( ˜D(j)) � +π′∈Σj ˜q((D−0)π′| ˜D(j)) +, +we have that +p := +m +� +i=0 +w˜q,Σ +D,i( ˜D(i))1[S( ˜D(i)) ≥ S(D)] +stochasically dominates the uniform distribution under H∝ +0 . +14We use the prefix “pseudo” since Σ need not be a group. +37 + +Proof sketch. We give a sketch of the proof as much of it is the same as that of Theorem 2.1. Using Bayes’ +theorem, we have that +P(D = di|( ˜D(0), . . . , ˜D(m)) ∈ pseudo-orbΣ(d)) += +P(( ˜D(0), . . . , ˜D(m)) ∈ pseudo-orbΣ(d)|D = di)f(di) +�m +j=0 P(( ˜D(0), . . . , ˜D(m)) ∈ pseudo-orbΣ(d)|D = dj)f(dj) += +f(di) � +π∈Σi P((D−0) = (dπ(1), . . . , dπ(m))|D = di) +�m +j=0 f(dj) � +π′∈Σj P((D−0) = (dπ′(1), . . . , dπ′(m))|D = dj). +Thus, conditional on pseudo-orbΣ(D), S(D)’s distribution is indeed the weighted distribution on the multiset +{S( ˜D) : ˜D ∈ D} with weight of the ith element equal to w˜q,Σ +D,i( ˜D(i)) under H∝ +0 . The rest of the proof, in this +setting of distinct resamples, follows in precisely the same manner as that of Theorem 2.1. +We now extend to the case of repeated samples via a coupling argument. Define D′ := D × [0 : m] so that, +for each point d in D it has m + 1 “clones” in D′ given by (d, 0), . . . , (d, m). We also define a density f ′ +on D′ given by f ′((d, i)) := (m + 1)−1f(d) for all d ∈ D, i ∈ [0 : m]. Lastly, we extend the test statistic S +to a test statistic S′ on D′ which also takes each clone to the value of its original: S′((d, i)) = S(d) for all +d ∈ D, i ∈ [0 : m]. +We will couple p to a random variable p′ obtained by a procedure on the space D′ which draws distinct +elements ˜D′(i), to be defined below. We call the original process (that may draw repeated elements and acts +only on D) P and the coupled process P′. The observed dataset that the coupled process P′ will see and +use is a cloned version (D, j) of the true observed data D, where j is sampled uniformly at random from the +set [0 : m]; importantly P′ treats this cloned dataset D′ := (D, j) as though it were the observed dataset. +Finally, we use ˜q′ to denote the conditional resampling distribution of P′, induced by ˜q. That is, for a list +D′ +−0 := ( ˜D′(1), . . . , ˜D′(m)) of resamples in D′, ˜q′(D′ +−0|D′) denotes the joint conditional probability of P′ +having obtained the list of resamples D′ +−0 given that it observed the dataset D′. Algorithm 2 describes how +P′ is defined with respect to P. +Algorithm 2: Coupling of P′ to P +Input: P′’s observed dataset D′ := (D, j) where j ∼ Unif([0 : m]) +1 ˜D′(0) ← D′ +2 D′(0) ← ( ˜D′(0)) +3 for i = 1, . . . , m do +4 +˜D(i) ← ith element of D in the process P +5 +j ← Unif({j′ ∈ [0 : m] : ( ˜D(i), j′) ̸∈ D′(i−1)}) +6 +˜D′(i) ← ( ˜D(i), j) +7 +D′(i) ← concatenation of D′(i−1) with ˜D′(i) +8 D′ ← D′(m) +Output: +p′ := +�m +i=0 1[S′( ˜D′(i)) ≥ S′(D′)]f ′( ˜D′(i)) � +π∈Σi ˜q′((D′ +−0)π| ˜D′(i)) +�m +j=0 f ′( ˜D′(j)) � +π′∈Σj ˜q′((D′ +−0)π′| ˜D′(i)) +, +At a high level, the coupled process P′ draws distinct elements by sampling an element’s clone number +j uniformly at random from all remaining yet-unselected values; using these distinct resamples, it then +calculates a p-value in precisely the same way as Algorithm 1 with weights w˜q,Σ +D′ ( ˜D′(i)). +Since the draws of P′ are guaranteed to be distinct, Lemma E.1 will guarantee p′’s stochastic domination +of the uniform distribution as long as f ′ is the true density of D′, the observed data for P′ (since ˜q′ is the +conditional resampling distribution and because f ′ is trivially proportional to itself, thereby satisfying the +proportionality hypothesis in this coupled model). For d′ ∈ D′, let C(d′) denote the second component of d′ +38 + +(i.e., the clone number of d′) and d be the first component (i.e., the uncloned version of d′ in D); then we +see that f ′ is D′’s true density: +P(D′ = d′) = P(C(D′) = C(d′)|D = d)P(D = d) += P(C(D′) = C(d′))P(D = d) +since the clone number j is sampled independently of D += (m + 1)−1f(d) +because j ∼ Unif([0 : m]) to clone D += f ′(d′) +as desired. +We complete the coupling by showing that +˜q′((d′ +−0)π|d′ +i) = κ˜q((d−0)π|di), ∀i ∈ [0 : m], π ∈ Σi for some κ > 0 depending on neither i nor π, +(14) +where d is some list (d0, . . . , dm) of elements in D and d′ := (d′ +0, . . . , d′ +m) is a list of elements of D′ comprising +distinct clones of the di (i.e., the first component of d′ +i is equal to di, and each element of d′ is distinct), +(d′ +−0)π := (d′ +π(1), . . . , d′ +π(m)), and (d−0)π := (dπ(1), . . . , dπ(m)). +To see why this is true, note that if P′ +observes d′ +i as the original dataset, then P must have seen its uncloned version: di. Thus, applying the +cloning number function C defined above to (d′ +−0)π elementwise, we have that +˜q′((d′ +−0)π|d′ +i) = P(D′ +−0 = (d′ +−0)π|D′ = d′ +i) += P(C(D′ +−0) = C((d′ +−0)π), D−0 = (d−0)π|C(D′) = C(d′ +i), D = di) += P(C(D′ +−0) = C((d′ +−0)π)|D−0 = (d−0)π, D′ = d′ +i)P(D−0 = (d−0)π|D = di) +as D−0 ⊥⊥ C(D′) | D . +To see that this is proportional to ˜q((d−0)π|di) (which is precisely the second term in the last line above), +let md((d−0)π) denote the number of elements in (d−0)π with first component equal to d, and observe that +we may write the first term in the last line above as +P(C(D′ +−0) = C((d′ +−0)π)|D−0 = (d−0)π, D′ = d′ +i) += +� +� +� +distinct d∈d not equal to di +1 +�md((d′ +−0)π) +k=1 +(m + 1 − k + 1) +� +� · +1 +�mdi((d′ +−0)π) +k=1 +(m + 1 − k) += +� +� +� +distinct d∈d not equal to di +(m + 1 − md((d′ +−0)π))! +(m + 1)! +� +� · (m − mdi((d′ +−0)π))! +m! +. +Now note that md((d′ +−0)π) does not depend on the choice of π ∈ Σi and so we need only consider the +permutation π∗ +i ∈ Σi given by +π∗ +i : k �→ +� +� +� +� +� +i if k = 0 +k − 1 if 0 < k ≤ i +k if i < k ≤ m +so that, for any π ∈ Σi, md((d′ +−0)π) = md((d′ +−0)π∗ +i ) = md(d′ +−i), where d′ +−i := (d′ +0, . . . , d′ +i−1, d′ +i+1, . . . , d′ +m). +As such, combining this fact with what has been shown above, gives that +P(C(D′ +−0) = C((d′ +−0)π∗ +i )|D−0 = (d−0)π∗ +i , D′ = d′ +i) = +� +� +� +distinct d∈d not equal to di +(m + 1 − md(d′ +−i))! +(m + 1)! +� +�·(m − mdi(d′ +−i))! +m! +. +Note that the right-hand side above depends on di only through its value, not its subscript, and hence the +equation above takes the same value for any j ̸= i for which dj = di. In the case of distinct i, j with di ̸= dj, +39 + +the above gives that +P(C(D′ +−0) = C((d′ +−0)π∗ +i )|D−0 = (d−0)π∗ +i , D′ = d′ +i) += +� +� +� +distinct d∈d equal to neither di nor dj +(m + 1 − md(d′ +−i))! +(m + 1)! +� +� · (m + 1 − mdj(d′ +−i))! +(m + 1)! +· (m − mdi(d′ +−i))! +m! += +� +� +� +distinct d∈d equal to neither di nor dj +(m + 1 − md(d′ +−j))! +(m + 1)! +� +� · (m + 1 − mdj(d′ +−i))! +(m + 1)! +· (m − mdi(d′ +−i))! +m! += +� +� +� +distinct d∈d equal to neither di nor dj +(m + 1 − md(d′ +−j))! +(m + 1)! +� +� · (m − mdj(d′ +−j))! +(m + 1)! +· (m + 1 − mdi(d′ +−j))! +m! += +� +� +� +distinct d∈d not equal to dj +(m + 1 − md(d′ +−j))! +(m + 1)! +� +� · (m − mdj(d′ +−j))! +m! += P(C(D′ +−0) = C(d′ +−j)|D−0 = d−j, D′ = d′ +j) += P(C(D′ +−0) = C((d′ +−0)π∗ +j )|D−0 = (d−0)π∗ +j , D′ = d′ +j), +as desired, where the third equality follows from the fact that mdj(d′ +−j)+1 = mdj(d′ +−i) for any pair i, j with +di ̸= dj. By the same argument made for π∗ +i , the last line of the above remains true if we replace π∗ +j with +any π ∈ Σj and thus, we have shown that not only does P(C(D′ +−0) = C((d′ +−0)π)|D−0 = (d−0)π, D′ = d′ +i) +not depend on the choice of π ∈ Σi, for any given i, but it also does not depend on i ∈ [0 : m], and hence we +have the desired proportionality of equation (14). +Finally, since f ′( ˜D′(i)) = (m+1)−1f( ˜D(i)) by definition, and ˆf( ˜D(i)) = Kf( ˜D(i)) ∀i ∈ [0 : m] for some K not +depending on i under the proportionality hypothesis, we have that ˆf( ˜D(i)) = K(m + 1)f ′( ˜D′(i)), ∀i ∈ [0 : m] +and hence the two are proportional. As such, we have that +�m +i=0 1[S′( ˜D′(i)) ≥ S′(D′)]f ′( ˜D′(i)) � +π∈Σi ˜q′((D′ +−0)π| ˜D′(i)) +�m +j=0 f ′( ˜D′(j)) � +π′∈Σj ˜q′((D′ +−0)π′| ˜D′(j)) += +�m +i=0 1[S( ˜D(i)) ≥ S(D)] ˆf( ˜D(i)) � +π∈Σi ˜q((D−0)π| ˜D(i)) +�m +j=0 ˆf( ˜D(j)) � +π′∈Σj ˜q((D−0)π′| ˜D(j)) +and so p′ = p. This combined with the fact shown above that p′ is a valid p-value implies the desired validity +of p under H∝ +0 . +E.2 +Sharing samples between y ∈ Y +We now show how the above framework can be used to share samples between different grid values y ∈ Y +as discussed in Remark 7. We describe this sample sharing in the absence of the rounding procedure of +Remark 7 which considers only a small grid Y′ ⊆ Y (i.e., we will obtain samples for each y in the full +support Y and share these samples amongst all other elements of Y), but both of these methods can be +combined to construct conformal prediction regions even more efficiently. +In more detail, recall that the naive interval construction described in Section 3.3 runs |Y| independent +non-stationarity tests, using our weighted MC randomization testing framework for conditionally i.i.d. re- +samples using q, at each y ∈ Y. +To formalize this notion, set (D[1:T −1], CT , XT ) to denote the entire +dataset except the last response and let Uy denote the exogenous randomness used by the weighted MC +randomization test when determining if y is in the acceptance region. Then we can define an acceptance +function ϕ(Uy, (D[1:T −1], CT , XT ), y) which is 1 (i.e., accepts) if the test, using Uy as exogenous random- +ness, states that y is in the acceptance region upon seeing (D[1:T −1], CT , XT ), and is otherwise 0 (i.e., +rejects). In the naive gridding procedure, the Uy are all jointly independent and are distributed identically +40 + +to the exogeneous randomness U that is used in the usual non-stationarity test. As such, it is clear that +E[ϕ(UYT , (D[1:T −1], CT , XT ), YT )] ≥ 1 − α due to the validity of the non-stationarity test proved in Sec- +tion 3.2 and so the corresponding prediction region attains coverage at least that of the nominal rate. The +Uy, however, need not be independent. Instead, for example, if we have Uy = U for all y ∈ Y, where, again, +U is the independent exogenous randomness used by the non-stationarity test, then our prediction region +would once again control miscoverage at the nominal rate. More generally, as long as each Uy (and U) is +generated independently from D and we have the following equality in distribution: +(UYT , YT ) +d= (U, YT ), +the resulting prediction regions will be valid. The process of sharing samples between y ∈ Y is based upon +this idea. +While described as resampling datasets ˜D(i) conditional on D in Section 3.2, the non-stationarity tests we +describe really sample permutations πi, conditionally on D, and then set ˜D(i) equal to Dπi := ((Cπi(1), +Xπi(1), Yπi(1)), . . . , (Cπi(T ), Xπi(T ), Yπi(T ))); we use qΠ to denote the corresponding (to q) resampling distri- +bution over permutations. In the naive interval construction procedure which uses independent exogenous +randomness, let Dy +Π denote the (multi)set of permutations sampled for determining if y is in the acceptance +region. We claim that all these samples can be shared so that each y ∈ Y instead uses the entire set of +shared permutations DΠ := � +y∈Y Dy +Π as opposed to just Dy +Π: +Proposition E.1. Let q be any conditionally i.i.d. resampling distribution and let qΠ denote the correspond- +ing resampling distribution over permutations. Furthermore define, for each y ∈ Y, a conditional distribution +q′ +Π,y over permutations which draws its samples conditionally independently from YT given (D[1:T −1], CT , XT ) +and is defined by +q′ +Π,y(·|((C1, X1, Y1), . . . , (CT , XT , YT ))) = qΠ (·|((C1, X1, Y1), . . . , (CT , XT , y))) . +To construct a prediction region, suppose that, for each y ∈ Y, a sampled (multi)set Dy +Π = {π1,y, . . . , πm,y} +of permutations is generated by sampling each permutation conditionally i.i.d. from q′ +Π,y(·|D)15, but the +full (multi)set DΠ of m|Y| permutations is used to determine y’s membership in the acceptance region. +Specifically, defining Dy := ((C1, X1, Y1), . . . , (CT , XT , y)), writing Y = {y1, . . . , y|Y|}, and defining Λ := +{(0, 0)} ∪ ([m] × [|Y|]), we construct an acceptance region A from DΠ via the rule: +y ∈ A ⇐⇒ +� +(i,j)∈Λ +wq,y +DΠ,i,yj((Dy)πi,yj )1[S((Dy)πi,yj ) ≥ S(Dy)] > α, +(15) +where +wq,y +DΠ,i,yj((Dy)πi,yj ) := +ˆf((Dy)πi,yj )q′ +Π,yj(π−1 +i,yj|(Dy)πi,yj ) +� +(˜i,˜j)∈Λ\{(i,j)} +q′ +Π,y˜j(π˜i,y˜j ◦ π−1 +i,yj|(Dy)πi,yj ) +� +(i′,j′)∈Λ +ˆf((Dy) +πi′,yj′ )q′ +Π,yj′ (π−1 +i′,yj′ |(Dy) +πi′,yj′ ) +� +(˜i′,˜j′)∈Λ\{(i′,j′)} +q′ +Π,y˜j′ (π˜i′,y˜j′ ◦ π−1 +i′,yj′ |(Dy) +πi′,yj′ ) +, +and we take π0,y0 to simply be the identity permutation. +Then the resultant prediction region A controls miscoverage at the nominal rate under the proportionality +hypothesis H∝ +0 . +Before giving a proof we make a brief computational/procedural note that, similar to equation (9) of Re- +15Since q′ +Π,y does not depend on YT , we can indeed sample from this distribution during prediction region construction +wherein YT is unobserved +41 + +mark 9, when q(·|(Dy)πi,yj) does not depend on (i, j) ∈ Λ for any y ∈ Y, we have that +wq,y +DΠ,i,yj((Dy)πi,yj ) = +ˆf((Dy)πi,yj )q′ +Π,yj(π−1 +i,yj|(Dy)πi,yj ) +� +(˜i,˜j)∈Λ\{(i,j)} +q′ +Π,y˜j(π˜i,y˜j ◦ π−1 +i,yj|(Dy)πi,yj ) +� +(i′,j′)∈Λ +ˆf((Dy) +πi′,yj′ )q′ +Π,yj′ (π−1 +i′,yj′ |(Dy) +πi′,yj′ ) +� +(˜i′,˜j′)∈Λ\{(i′,j′)} +q′ +Π,y˜j′ (π˜i′,y˜j′ ◦ π−1 +i′,yj′ |(Dy) +πi′,yj′ ) += +ˆf((Dy)πi,yj )q′ +Π,yj(π−1 +i,yj|(Dy)πi,yj )/q′ +Π,yj(πi,yj ◦ π−1 +i,yj|(Dy)πi,yj ) +� +(i′,j′)∈Λ +ˆf((Dy) +πi′,yj′ )q′ +Π,yj′ (π−1 +i′,yj′ |(Dy) +πi′,yj′ )q′ +Π,yj′ (πi′,yj′ ◦ π−1 +i′,yj′ |(Dy) +πi′,yj′ ) += +ˆf((Dy)πi,yj )q′ +Π,yj(π−1 +i,yj|(Dy)πi,yj )/q′ +Π,yj(id|(Dy)πi,yj ) +� +(i′,j′)∈Λ +ˆf((Dy) +πi′,yj′ )q′ +Π,yj′ (π−1 +i′,yj′ |(Dy) +πi′,yj′ )q′ +Π,yj′ (id|(Dy) +πi′,yj′ ) +, +where id denotes the identity permutation. Hence, computation of wq,y +DΠ,i,yj((Dy)πi,yj ) across all (i, j) ∈ Λ +becomes tractable in only O(m|Y|) operations. +Proof. Consider a non-stationarity test in which we draw m|Y| samples and the resampling distribution ˘q +we consider ignores YT , and instead draws these m|Y| samples in |Y| groups, the jth of which consists of m +resamples Dπ1,yj , . . . , Dπm,yj where the πi,yj are drawn conditionally i.i.d. from q′ +Π,yj (·|D) for each j ∈ [|Y|], +and where Y = {y1, . . . , y|Y|}. It is important to note that ˘q is not a conditionally i.i.d resampling scheme +and thus, as discussed in Section 5.3.2, the workaround discussed in Section E.1 must be employed. Defining +Σ = {(0 ℓ) : ℓ ∈ [0 : m|Y|]} and letting i ∈ [0 : m] index with any given group and j ∈ +� +[|Y|] if i > 0 +{0} if i = 0 +index over groups, notice that +wq,YT +DΠ,i,yj(Dπi,yj ) = w˘q,Σ +D,(i,j)( ˜D(i,j)), +where D denotes the full set of m|Y| resamples and we double index resamples as D(i,j) = Dπi,yj for all +(i, j) in Λ. This is because, letting q′ +yj denote the induced (by q′ +Π,yj) distribution on each resample given +by q′ +yj(Dπ|D) = q′ +Π,yj(π|D) and looking at the rightmost terms in the numerator of wq,y +DΠ,i,yj((Dy)πi,yj )’s +definition, we have that +q′ +Π,yj(π−1 +i,yj|Dπi,yj ) +� +(˜i,˜j)∈Λ\{(i,j)} +q′ +Π,y˜j(π˜i,y˜j ◦ π−1 +i,yj|Dπi,yj ) += q′ +yj(D|Dπi,yj ) +� +(˜i,˜j)∈Λ\{(i,j)} +q′ +y˜j(D +π˜i,y˜j |Dπi,yj ) += q′ +yj(D| ˜D(i,j)) +� +(˜i,˜j)∈Λ\{(i,j)} +q′ +y˜j( ˜D(˜i,˜j)| ˜D(i,j)) += +� +π∈Σ(i,j) +˘q((( ˜D(0,0), ˜D(1,1), . . . , ˜D(m,1), . . . , ˜D(1,|Y|), . . . , ˜D(m,|Y|))π)−0| ˜D(i,j)), +where Σ(i,j) is the single element subset of Σ containing solely the permutation on [0 : m|Y|] that swaps 0 +and m(j − 1) + i. Thus, the numerators of wq,YT +DΠ,i,yj(Dπi,yj ) and w˘q,Σ +D,(i,j)( ˜D(i,j)) are equal, and precisely the +same argument shows that the denominators are also equal and thus wq,YT +DΠ,i,yj(Dπi,yj ) = w˘q,Σ +D,(i,j)( ˜D(i,j)). +Hence, the p-value +� +(i,j)∈Λ +wq,y +DΠ,i,yj(Dπi,yj )1[S(Dπi,yj ) ≥ S(D)], +corresponding to line (15) is a valid p-value by the main result of Section E.1. +Letting U denote the +exogenous randomness used by this test, notice that the exogenous random variables Uy used by each y ∈ Y +42 + +in the prediction region construction of line (15) are all (deterministically) equal—since the only randomness +in determining if y ∈ A is in the random sampling of permutations DΠ, which does not depend on the +specific choice of y ∈ Y—and marginally identically distributed to U. Since each of U and Uy is generated +independently from YT , we have that (UYT , YT ) +d= (U, YT ) and hence the corresponding prediction region A +is valid, as desired. +F +Pseudocode for resampling distributions +In this appendix section, we provide pseudocode for all resampling procedures described in Section 4. +Throughout this section, we use the notation Cat(p), for any p ∈ [0, 1]d with �d +i=1 pi = 1, to denote +the categorical distribution on [d] with probability of sampling i equal to pi. +F.1 +Non-stationarity testing in a C-stationary strongly non-reactive environ- +ment +We begin with the resampling procedures for non-stationarity testing in a C-stationary strongly non-reactive +environment (Environment 5). +Algorithm 3: imitationπ +Input: Data sequence D +1 Set ˜D to the empty list and set R ← [T] +2 for t = 1, . . . , T do +3 +Sample +i ∼ Cat +� +� +� +PA(Xj| ˜D, Cj)1[j ∈ R] +�T +j′=1 PA(Xj′| ˜D, Cj′)1[j′ ∈ R] +�T +j=1 +� +� , +if PA(·| ˜D)’s support intersects R; otherwise, terminate the sampling procedure +4 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +Algorithm 4: re-imitationπ +Input: Data sequence D; probability distribution PUt denoting the distribution of the tth exogenous +random variable Ut generated by A +1 Set ˜D to the empty list and set R ← [T] +2 for t = 1, . . . , T do +3 +Sample +˜Ut ∼ PUt(·|∃s ∈ R : Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut)) +if the conditioning event is non-empty; otherwise, terminate the sampling procedure +4 +Sample i ∼ Unif +� +{s ∈ R : Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut)} +� +5 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +43 + +Algorithm 5: cond-imitationπ +Input: Data sequence D as well as the exogenous randomness U1, . . . , UT used to generate it +1 Set ˜D to the empty list and set R ← [T] +2 for t = 1, . . . , T do +3 +Sample i ∼ Unif +� +{s ∈ R : Xs = δt(Cs, ˜Ht−1, U1, . . . , Ut)} +� +if the set is non-empty; otherwise, +terminate the sampling procedure +4 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +F.2 +Non-stationarity testing in an MDP +As discussed in Section 4.2, the datasets used in non-stationarity testing in an MDP are augmented with +an additional action XT +1, and thus we may view these datasets as a list of T + 1 state action pairs: +D = ((C1, X1), . . . , (CT +1, XT +1)). Under this framework, all resampling procedures in this section utilize +the following function φ, which takes the state-action pair (c, x) to the set of indices which follow it: +φ(c, x) = {t ∈ [2 : T + 1] : (Ct−1, Xt−1) = (c, x)}. +Additionally, using this view of the dataset D, we use Zt to denote the tth state-action pair (Ct, Xt); ˜Zt +denotes the tth state-action pair of the resampled dataset ˜D. Using the function φ, we now present the +corresponding uniformπ, imitationπ, re-imitationπ, and cond-imitationπ for an MDP under this setup of the +dataset D. +Algorithm 6: uniformπ in an MDP +Input: Data sequence D +1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] +2 for t = 2, . . . , T + 1 do +3 +Sample +i ∼ Unif +� +R ∩ φ( ˜Zt−1) +� +if the set is non-empty; otherwise, terminate the sampling procedure +4 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +Algorithm 7: imitationπ in an MDP +Input: Data sequence D +1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] +2 for t = 2, . . . , T + 1 do +3 +Sample +i ∼ Cat +� +� +� +PA(Xj| ˜D, Cj)1[j ∈ R ∩ φ( ˜Zt−1)] +�T +1 +j′=2 PA(Xj′| ˜D, Cj′)1[j′ ∈ R ∩ φ( ˜Zt−1)] +�T +1 +j=2 +� +� , +if ∃j ∈ R ∩ φ( ˜Zt−1) such that PA(Xj| ˜D, Cj) > 0; otherwise, terminate the sampling procedure +4 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +44 + +Algorithm 8: re-imitationπ in an MDP +Input: Data sequence D; probability distribution PUt denoting the distribution of the tth exogenous +random variable Ut generated by A +1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] +2 for t = 2, . . . , T + 1 do +3 +Sample +˜Ut ∼ PUt(·|∃s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut)) +if the conditioning event is non-empty; otherwise, terminate the sampling procedure +4 +Sample i ∼ Unif +� +{s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, ˜U1, . . . , ˜Ut)} +� +5 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +Algorithm 9: cond-imitationπ in an MDP +Input: Data sequence D as well as the exogenous randomness U1, . . . , UT +1 used to generate it +1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] +2 for t = 2, . . . , T + 1 do +3 +Sample i ∼ Unif +� +{s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, U1, . . . , Ut)} +� +if the set is non-empty; +otherwise, terminate the sampling procedure +4 +Append Zi to ˜D and set R ← R\{i} +Output: ˜D +F.3 +Conditional independence testing +We now give pseudocode for our resampling procedures for conditional independence testing discussed in +Section 4.3. We first show pseudocode for restricted-uniformπ resampling, which, although not used on its +own for conditional independence testing, makes up the first stage of the restricted-uniformπ+imitationX +resampling scheme. We then go on to present the imitationX resampling procedure, which is also a key ingre- +dient that is used in the uniformπ+imitationX and restricted-uniformπ+imitationX resampling procedures, +which are presented subsequently. Finally, we show pseudocode for combinedπ,X sampling, which combines +the permutation and randomization of Xt’s into a single stage. +Algorithm 10: restricted-uniformπ +Input: Data sequence D +1 Set ˜D to the empty list +2 Set Γ ← {π′ ∈ Π[T ] : g(Xπ′(t)) = g(Xt), ∀t ∈ [T]} +3 Sample π ∼ Unif(Γ) +4 ˜D ← (Zπ(t))T +t=1 +Output: ˜D +G +Supplementary simulation results +G.1 +MCMC plots +In this section, we present the power, Type-I error, coverage, and length plots for the unweighted MCMC +randomization test and its inversion, for the inferential tasks discussed in Section 5, in Figures 9–14. +45 + +Algorithm 11: imitationX +Input: Data sequence D +1 Set ˜D to the empty list +2 for t = 1, . . . , T do +3 +Sample +˜Xt ∼ PA(·| ˜D, Ct, g(Xt)), +if there exists x ∈ X with PA(x| ˜D, Ct, g(Xt)) > 0; otherwise, terminate the sampling procedure +4 +Append ˜Zt := (Ct, ˜Xt, Yt) to ˜D +Output: ˜D +Algorithm 12: uniformπ+imitationX +Input: Data sequence D +1 Sample D′ according to the uniformπ distribution applied to D +2 Sample ˜D according to the imitationX distribution (Algorithm 11) applied to D′ +Output: ˜D +Algorithm 13: restricted-uniformπ+imitationX +Input: Data sequence D +1 Sample D′ according to the restricted-uniformπ distribution (Algorithm 10) applied to D +2 Sample ˜D according to the imitationX distribution (Algorithm 11) applied to D′ +Output: ˜D +Algorithm 14: combinedπ,X +Input: Data sequence D +1 Set ˜D to the empty list and set R ← [T] +2 for t = 1, . . . , T do +3 +Sample +i ∼ Cat +� +� +� +� +� +� +x∈X:g(x)=g(Xj) PA(x| ˜D, Cj)1[j ∈ R] +�T +j′=1 +� +x′∈X:g(x′)=g(Xj′) PA(x′| ˜D, Cj′)1[j′ ∈ R] +� +� +T +j=1 +� +� +� , +if there exists j ∈ R such that � +x∈X:g(x)=g(Xj) PA(x| ˜D, Cj) > 0; otherwise, terminate the +sampling procedure +4 +Set R ← R\{i} +5 +Sample +˜Xt ∼ PA(·| ˜D, Ci, g(Xi)), +if there exists x ∈ X with PA(x| ˜D, Ci, g(Xi)) > 0; otherwise, terminate the sampling procedure +6 +Append ˜Zi := (Ci, ˜Xt, Yi) to ˜D +Output: ˜D +G.2 +Computation times +In this section, we plot the computation time curves (to compute p) for all resampling algorithms, environ- +ments, adaptive assignment algorithms, and types of randomization test (i.e., weighted MC or unweighted +MCMC) discussed in Section 5 in Figures 15–22. +46 + +Figure 9: Type-I error rate (leftmost) and power (second from left) of both weighted MC and unweighted MCMC +randomization tests at fixed m = 100 and varying T as well as power at fixed T = 100 and varying m (third from +left) and fractional effective sample size plots at fixed m = 100 and varying T (rightmost) in a contextless stationary +strongly non-reactive environment on data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +Figure 10: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC +randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from +right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary +strongly non-reactive environment with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +G.3 +Auxiliary simulation results +In this section we present all remaining auxiliary simulation results. Figure 23 displays the power plot at +fixed T = 100 and varying m for the conditional independence test in the contextual stationary strongly +non-reactive environment on data gathered via ϵ-greedy and LinUCB discussed in Section 5.1.1. On the other +hand, Figure 24 illustrates that the phenomenon of shift in relative performance of the uniform i.i.d. baseline +in comparison to both ϵ-greedy and LinUCB, described in Section 5.2.2 also occurs when the baseline is com- +pared to a biased i.i.d. adaptive assignment algorithm which selects actions at each timestep independently +from 2Bern(0.1) − 1. +47 + +Contextless C-stationary strongly non-reactive environment, non-stationanity test +Type-I error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 +0.8 +Power +0.8 +0.8 +Power +0.100 +Type-l +0.6 +0.6 +0.6 +0.075 +Average +0.4 +0.4 +0.4 +0.050 +0.2 +0.2 +0.2 +0.025 +0.000 +0.0 +75 +0.0 +0.0 +25 +50 +75 +100 +25 +50 +100 +102 +103 +104 +25 +50 +75 +100 +T +T +m +T +-+- uniform id baseline +-greedy cond-imitationn, MC +-greedy uniformn, MCMC +-greedycond-imitationn,MCMC ++-greedyuniform,MC +.....UCB uniformn, MC +-greedyimitationr,MCMC +....UCB uniformn,MCMC ++-greedyimitationr,MC +UCBimitationr,MC +-greedy re-imitationr, MCMC +*..UCBimitationn,MCMC +-greedy re-imitationn, MCContextless stationary strongly non-reactive environment conditional independence test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 + 0.125 +0.8 +0.8 +.... +0.8 +Power +0.100 +mativ +Pow +pe +0.6 +0.6 +0.6 +0.075 +Average +Average +0.4 +0.4 +0.4 +0.050 +I 0.025 +0.2 +0.2 +0.2 +0.000 +0.0 +0.0 +25 +0.0 +25 +50 +75 +100 +50 +75 +100 +102 +103 +104 +25 +75 +100 +T +T +m +T ++- uniform id baseline +UCB restricted-uniformn+imitationx, MC +-greedy uniform+imitationx,MCMC +-greedyrestricted-uniform,+imitationx,MC +UCB uniformn+imitationx, MC +*..UCB combinedn.,MCMC +-greedy combinedn,x, MC +-greedyrestricted-uniform+imitationx,MCMC +..-*..UCB restricted-uniformn+imitationx, MCMC +--greedyuniform,+imitationx,MC +* -greedy combinedn,x,MCMC +..UCBuniform,+imitationx,MCMC +.....UCB combinedn.x, MCFigure 11: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC +randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from +right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary +strongly non-reactive environment with data gathered via ϵ-greedy, LinUCB, and the uniform i.i.d. baseline. +Figure 12: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC +randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from +right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary +strongly non-reactive environment with data gathered via ϵ-greedy and greedy Q-learning. +48 + +Contextual C-stationary strongly non-reactive environment, non-stationarity test +Type-I error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 +1.0 +1.0 - +Power +0.8 +Power +0.8 +0.100 +Type-l +0.6 +0.6 +0.6 +0.075 +Average +0.4 +0.4 +0.4 +0.050 +0.2 +0.2 +0.2 +0.025 +0.000 +0.0 +0.0 +0.0 +25 +50 +75 +100 +25 +50 +75 +100 +102 +103 +104 +25 +50 +75 +100 +T +T +m +T +-+- uniform iid baseline +-greedy cond-imitationn, MC +-greedy uniformn, MCMC +*-greedycond-imitationn,MCMC +-greedyuniformn,MC +.....LinUCB uniformn,MC +-greedyimitationr,MCMC +..... LinUCB uniformn, MCMC ++-greedyimitationu,MC +LinUCBimitationr,MC +-greedyre-imitationn,MCMC +*..LinUCBimitationn,MCMC +-greedy re-imitationn,MCMDP, non-stationarity test +Type-l error +Power (fixed m) +Power (fixed T) +Fractional effective sample size +0.150 +1.0 - +1.0 +1.0 +0.8 +0.8 +0.100 +Power +0.6 +0.6 +0.6 +Average +Average +0.4 +0.4 +0.4 +0.050 + 0.025 +0.2 +0.2 +0.2 +0.000 +0.0 +25 +0.0 +0.0 +25 +50 +75 +100 +50 +75 +100 +102 +103 +104 +25 +75 +100 +T +T +m +T +-+- uniform iid baseline +.... greedy uniformr, MC +* -greedy re-imitationn,MCMC +--greedyuniformr,MC +greedy imitation,MC +*-greedy cond-imitation,MCMC +-greedyimitationr,MC +*-greedyuniformn,MCMC +.....greedy uniformr,MCMC +--greedyre-imitation,MC +*-greedyimitationn,MCMC +.. +greedy imitationn, MCMC +-greedy cond-imitationn, MCFigure 13: Coverage and average length of confidence intervals for b0 using both weighted MC and unweighted MCMC +randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +Figure 14: Coverage and average length of conformal prediction intervals for YT using both weighted MC and +unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +49 + +Confidence interval +1.00 +0.98 +0.96 +Average Length +6 +Cove +0.92 +eja +0.90 +0.88 +0.86 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +T +T +uniformiid comparator +UCB restricted-uniform+imitationx,MC +-greedyuniformn+imitationx,MCMC +-greedy restricted-uniformn+imitationx, MC +UCB uniformn+imitationx,MC +..UCBcombinedn.x,MCMC +-greedy combinedn,x,MC +-greedy restricted-uniform,+imitationx,MCMC +.....UCB restricted-uniform,+imitationx,MCMC +-greedyuniformn+imitationx,MC +一-greedycombinedn,x,MCMC +UCB uniformn+imitationx,MCMC +.... UCB combinedn.x, MCContextless C-stationary strongly non-reactive environment, conformal prediction interval +1.00 +10 +0.98. +8 +0.96 +abe. +Average Length +0.94 +0.90 +0.88 +0.86 +20 +60 +LQ +T +40 +8 +100 ++-uniformidbaseline ++-greedy cond-imitationr,MC +-greedyuniformn,MCMC +*-greedycond-imitationn,MCMC ++-greedyuniform,MC +....UCBuniformnMC +E-greedy imitationn,MCMC +....UCB uniformn,MCMC +--greedyimitationn,MC +....UCBimitation..MC +-greedy re-imitationr,MCMC +...UCB imitationn,MCMC ++-greedyre-imitationn,MCFigure 15: Computation times under the null (left) and alternative (right) distributions of randomization tests at +fixed m = 100 and varying T in a contextual stationary strongly non-reactive environment on data gathered via +ϵ-greedy and LinUCB. Note that the computation times for LinUCB imitationX are omitted as both Type-I error +and power curves are simply generated i.i.d. Bern(0.05). +Figure 16: Computation times under the null (left) and alternative (right) distributions of both weighted MC and +unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless stationary strongly non- +reactive environment on data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +50 + +Contextual stationary strongly non-reactive environment conditional independence test. computation times +14 +14 +12 +12 +(seconds) +AverageTime (seconds) +10 +10 +8 +8 +6 +6 +4 +N +20 +40 +60 +80 +40 +Q9 +80 +100 +20 +T +T +uniformiidbaseline +E-greedyuniform+imitationx +E-greedyimitationx(priorwork) +LinUcB uniform+imitationxNull distribution +Alternative distribution +5 +2 +20 +40 +Q9 +80 +100 +20 +40 +09 +80 +100 +T +T ++-uniformidbaseline +UCB restricted-uniform.+imitationx,MC +-greedyuniformn+imitationx,MCMC +-greedyrestricted-uniform+imitationx,MC +UCB uniformn+imitationx,MC +....UCB combineds.x +- -greedy combinedn,x, MC +-greedy restricted-uniformn+imitationx,MCMC +-*..UCBrestricted-uniform+imitationx,MCMC +-greedyuniform,+imitationx,MC + ε-greedy combinedn,x, MCMC +UCB uniformn+imitationx, MCMC +......UCB combinedn. Figure 17: Computation times under the null (left) and alternative (right) distributions of both weighted MC and +unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly +non-reactive environment with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +Figure 18: Computation times under the null (left) and alternative (right) distributions of both weighted MC and +unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly +non-reactive environment with data gathered via ϵ-greedy, LinUCB, and the uniform i.i.d. baseline. +51 + +Contextless C-stationary strongly non-reactive environment, non-stationarity test, average computation times +Null distribution +Alternative distribution +QT +10 +8. +8 +Average Time (seconds) +6 +4 +2 +2 +20 +40 +09 +08 +100 +20 +40 +09 +80 +100 +T +T ++- uniformiid baseline +-greedy cond-imitationn,M +*-greedyuniformn,MCMC +*-greedy cond-imitationn,MCMc +-greedyuniformn,MC +.UCBuniform,MC +--greedyimitationn,MCMC +....UCBuniformnMCMC +-greedyimitationr,MC +.....UCB imitationn,MC +-greedy re-imitationn,MCMC +.....UCBimitationn,MCMC +-greedy re-imitationn, MCContextual c-stationary strontly non-reactive environment, non-stationarity test, computation times +Null distribution +Alternative distribution +300 +DCE +250 +250 +(seconds) +200 +200 +150 +150 +abe. +100 +JaAY +100 +50 +[0] +0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +T +T ++- uniform iid baseline +-greedy cond-imitationn,M +*-greedyuniformr,MCMC +-greedycond-imitationn,MCMc +-greedyuniformn,MC +...LinUCB uniformn, MC +-greedyimitationn,MCMC +......LinUCB uniformn,.MCMC +-greedyimitation,MC +.....LinUCB imitationn,MC +E-greedyre-imitationn,McMc +.....LinUCB imitationn,MCMC +-greedy re-imitationn, MCFigure 19: Computation times under the null (left) and alternative (right) distributions of both weighted MC and +unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly +non-reactive environment with data gathered via ϵ-greedy and greedy Q-learning. +Figure 20: Computation time of construction of confidence interval for b0 using both weighted MC and unweighted +MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +52 + +MDP, non-stationarity test, computation times +Null distribution +Alternative distribution +14 +14 +12 +12 + (seconds) +Average Time (seconds) +10 +10 +8 +8 +6 +6 +4 +2 +N +0 +0 +20 +40 +60 +80 +100 +20 +40 +60 +08 +100 +T +T +-+- uniform iid baseline +.....greedy uniformn, MC +* -greedy re-imitationn,MCMC +--greedyuniformn,MC +..greedyimitationMC +-greedycond-imitation,MCMC +-greedyimitationr,MC +*-greedyuniformn,MCMC +....greedy uniformn,MCMC +--greedyre-imitationn,MC +* -greedy imitationn,MCMC +..*..greedy imitationn,MCMC +-greedycond-imitationn,MCConfidence interval, computation times +45 +40 +35 +CE +25 +20 +15 +10 +5. +0 +20 +40 +60 +80 +100 +T ++ +uniformiid comparator +UCB restricted-uniform,+imitationx,MC +-greedy uniform,+imitationx,MCMC +-greedy restricted-uniformn+imitationx,MC +UCB uniformn+imitationx,MC +.... UCB combinedn.x. +-greedy combinedn,x, MC +-greedy restricted-uniform,+imitationx,MCMC +..*.. UCB restricted-uniformn+imitationx, MCMC +-greedyuniformn+imitationx,MC + -greedy combinedr,x,MCMC +UCB uniformn+imitationx, MCMC +.....UCB combinedn.xFigure 21: Computation time of construction of conformal prediction interval for YT using both weighted MC and +unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +Figure 22: Computation time of construction of conformal prediction interval for YT using the MC randomization +test with data gathered via ϵ-greedy, UCB, and the uniform i.i.d. baseline. +53 + +Contextless C-stationary strongly non-reactive environment, conformal prediction interval, computation times +70 +Q9 +50 +40 +20 +10 +20 +40 +09 +08 +100 +T ++-uniformiidbaseline +...UCBimitationn.MC +-greedyimitationr,MCMC +.....UCBimitation,MCMC ++-greedyimitationn,MC ++ -greedy uniformn,MC +-greedy re-imitationn, MCMC +*一-greedyuniformn,MCMC +- -greedy re-imitationn,MC +.....UCBuniformuMC +-greedycond-imitationn,MCMC +...UCBuniform,MCMC +-greedy cond-imitationn,MCContextual C-stationary strongly non-reactive environment, conformal prediction interval, shared samples, computation times +m = 10 Mc samples +m =1oo Mc samples +50 +8 +Average Time (seconds) +5 +CE +20 +2 +10 +0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +T +人 +uniform iid baseline +-greedy re-imitationr +..UCBimitationn +...UCBuniform. +-greedy imitationr +-greedycond-imitation +E-greedy uniformmFigure 23: Power of randomization tests at fixed T = 100 and varying m in a contextual stationary strongly non- +reactive environment on data gathered via ϵ-greedy and LinUCB. +Figure 24: Power comparison of uniform i.i.d. baseline versus biased i.i.d. baseline that selects actions independently +from 2Bern(0.1) − 1 +54 + +Contextual stationary strongly non-reactive environment conditional independence test +1.0 +0.8 +Average Power +0.6 +0.4 +0.2 +0.0 +102 +103 +104 +m +-greedy imitationx (priorwork) +LinUCB uniformn+ imitationx +E-greedyuniformn+imitationx +LinUCB imitationx (priorwork)Contextual c-stationary strongly non-reactive environment, non-stationarity test, comparison of lid assignments: power +1.0 +0.8 +0.6 +Po +abejai +0.4 +0.2 +0.0 +20 +40 +09 +80 +100 +uniform iid baseline +biased iid \ No newline at end of file diff --git a/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/load_file.txt b/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..360f35de7af38dea4c9a78514380810d838c2457 --- /dev/null +++ b/QNE4T4oBgHgl3EQf-w4H/content/tmp_files/load_file.txt @@ -0,0 +1,2712 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf,len=2711 +page_content='Randomization Tests for Adaptively Collected Data Yash Nair and Lucas Janson Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Harvard University Abstract Randomization tests (including permutation tests) are one of the most fundamental methods in statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' enabling a range of inferential tasks such as testing for (conditional) independence of random variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' constructing confidence intervals in semiparametric location models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and constructing (by inverting a permutation test) model-free prediction intervals via conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Randomization tests are intuitive, easy to implement, and exactly valid for any sample size, but their use is generally confined to independent and/or exchangeable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Yet in many applications including clinical trials, online education, online advertising, and protein design, data is routinely collected adaptively, meaning that the aspects of the data under the data collector’s control (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', treatment assignments) are assigned at each time step via a (possibly randomized) algorithm that depends on all the data observed so far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' such assignment algorithms include (contextual) bandit and reinforcement learning algorithms as well as adaptive experimental designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this paper we present a general framework for randomization testing on adaptively collected data (despite its non-exchangeability), encompassing (and in some cases improving) the few existing results on randomization testing and conformal inference for adaptively collected data, as well as many other important settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The key to our framework is the ability to compute likelihood- ratio-based weights involving known quantities based purely on the known adaptive assignment algorithm, as long as a certain proportionality condition is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' These weights can then be accounted for in our framework to conduct an exact randomization test, but in order for the test to be powerful, resamples need to be diverse yet have weights as close to equal as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, we additionally present novel computationally tractable resampling algorithms for various popular adaptive assignment algorithms, data-generating environments, and types of inferential tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we demonstrate via a range of simulations our framework’s power (in the case of hypothesis testing) and narrow widths (in the case of confidence or prediction intervals produced by inverting randomization tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Motivation Randomization tests form an important methodological framework that enjoys a wide array of uses across statistics, ranging from testing equality of distributions to testing (conditional) independence between co- variates and response in supervised learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Beyond these classical uses of randomization tests, the framework also encompasses permutation testing and (inversely) conformal inference (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We briefly note here that, throughout this paper, we use the phrase “randomization test” not to refer to a test applied on data obtained via the physical act of randomization taken by an experimenter, but rather the randomization used by the test itself to compute a p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, we view a randomization test as a randomized procedure, taking the observed data set D as input, that samples m copies ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) that are jointly exchangeable with D under the null, and then computes a (valid) p-value by taking a simple average of indicators comparing the resampled data to the observed data via a given test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This interpretation of a randomization test has been referred to as a quasi-randomization test by Zhang and Zhao (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While randomization tests can make very weak assumptions on the marginal distribution of observations, they generally make quite restrictive assumptions on the joint distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05365v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='ME] 13 Jan 2023 In particular, they usually require independent and/or exchangeable data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 1937;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Pitman, 1937;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Lehmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Edgington and Onghena, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Candes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018), an assumption that is often violated in many real-world settings in which data are gathered in an adaptive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One such example is drug discovery research, in which a scientist adaptively submits experiments serially, where the tth experiment’s design is based upon the results of the first t − 1 (Popova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This complex dependency will, in general, violate the exchangeability assumption when analyzing the data from all experiments at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Similar situations arise in experimental designs in the natural sciences more broadly in addition to mobile health, adaptive clinical trials, online education, and online advertising, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Scenarios like those above naturally lead to a number of important inferential tasks that need to be performed on adaptively collected data, all of which we show can be performed via a randomization testing-based framework: Testing if two or more treatments induce the same distribution over outcomes Detecting non-stationarity in the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' testing whether the outcome depends not only on the treatment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' but also on time) Using past data to predict the outcomes of future samples with quantifiable uncertainty Constructing confidence intervals for the difference in locations of outcome distributions corresponding to different treatments in semiparametric models We now briefly describe various settings in which the above tasks and generalizations thereof are both difficult and important problems and motivate them with real-world examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Problem Domain 1 (Comparing response distributions of different covariates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' An interesting and chal- lenging problem in adaptive data collection settings is to test whether two or more different treatments amongst a total of K give rise to the same response distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', for some pair i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , K}, is (Y | X = i) d= (Y | X = j)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the most extreme case, one might ask if there is even any dependency at all between X and Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', whether the treatment has any effect on the outcome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Beyond the mobile health examples delineated below in Problem Domain 2, for which it is important to test if two different treatments actually have the same effect (or whether treatments have any effect at all), such testing is also useful in recommender systems (Schafer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2010), to determine if different content dis- plays actually produce different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Further complicating this setting is that different users may react differently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', as measured by clickthrough rate) to different content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, one might ask: Given the ‘context’ surrounding the user’s preferences, do they react the same way to different advertisements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Problem Domain 2 (Testing non-stationarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' An important problem that arises in domains in which actions are adaptively chosen and outcomes observed, is that of non-stationarity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, one may ask if the conditional distributions Yt | Xt in data gathered via some adaptive decision-making procedure change with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One such scenario in which non-stationarity testing is important is in mobile health applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For example, the Just-in-Time Adaptive Intervention (JITAI) (Nahum-Shani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2016) is a mobile health framework designed to provide appropriate support to patients with dynamic, time-varying states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', mood, location, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' JITAIs have been applied to, for instance, suicide prevention (Coppersmith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021), smoking relapse prevention (Battalio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021), and addiction science (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In situations like these, it is vital to be able to know whether or not a patient’s response, Yt, to the treatment, Xt, is varying over time, t, so as to facilitate administration of the most effective and appropriate care possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Tests of non-stationarity provide a principled method of achieving this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Problem Domain 3 (Predicting with high confidence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In experimental design settings, providing prediction intervals for the results of future experiments can be instrumental in guiding researchers in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A motivating example comes from Alvarsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In their work, the authors introduce conformal inference (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Shafer and Vovk, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Vovk, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Burnaev and Vovk, 2014) to researchers in the field of drug discovery and go on to give an example of how the methodology—which 2 also assumes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' or exchangeable data—can be applied to classify various types of ATP-Binding Cassette transporters at any user-specified uncertainty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', significance) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, in light of the sequential and adaptive nature of many experimental designs in this field and in related fields like drug development (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Mahajan and Gupta, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Godfrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Pallmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018), there is need for provably valid test-time prediction intervals for these adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Problem Domain 4 (Relating parameters in semiparametric models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A final difficult problem in scenarios like those of the previous Problem Domains is to estimate how the parameters of different response distribu- tions belonging to a semiparametric model relate to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For example, suppose that Y | X = x ∼ pθx, where {pθ : θ ∈ Rd} is a location family with parameter indexed by the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' How can we estimate θx − θx′ for treatments x and x′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This problem setting is ubiquitous, for example, in the multi-armed bandit literature, where it is common to assume that the reward distributions of all arms are distributed accord- ing to a location family like the Normal (Lai and Robbins, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such settings, being able to estimate parametric relationships between different treatments in finite samples is useful for a variety of real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As an example, as described in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020), performing post hoc inference can be useful in many settings ranging from recommender systems (Mary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2015) to anomaly detection (Ban and He, 2020) to finance and portfolio selection (Huo and Fu, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such settings, after-the-fact inference may be helpful to researchers and practitioners who wish to understand the differences between different treatments and potentially utilize such differences to guide future decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A number of tasks that we consider in this paper—and, indeed, some of those presented in the Problem Domains above—involve the construction of prediction or confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, as we show, con- structing such intervals reduces to hypothesis testing by inverting the corresponding test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, although not usually framed this way, conformal prediction is the inversion of a permutation test, a specific type of randomization test (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More generally, although it is unusual to invert a randomization test, we show that by applying certain transformations to the data, our randomization tests can be inverted to construct confidence intervals in semiparametric models (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Rabideau and Wang (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 delineates precisely how our tests can be inverted to produce conformal/confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Due to this inverse relationship between prediction/estimation and testing, we generally (with the exception of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3) restrict our attention solely to hypothesis testing for clarity of presentation, and only state results in terms of the validity of such tests (and, in particular, the stochastic domination of the uniform distribution by their p-values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, all of the problem domains which motivate this work are centered around adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Indeed, as we show in this paper, we will be able to answer all of these inferential questions on adaptively collected data via a general framework for randomization testing as long as the adaptivity is known to the analyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, we assume that the analyst has access to the (potentially non-deterministic) decision- making algorithm used to assign treatments in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now introduce some notation to define this setup and formalize these notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Notation The adaptive data collection settings we consider in this paper—of which popular settings like bandits, Markov Decision Processes (MDPs), reinforcement learning, and adaptive experimental designs are all special cases—comprise a data collection horizon of T, the number of timesteps of data collected by the practitioner (we will treat T as deterministic in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' At each timestep t, a context (sometimes called a state) Ct ∈ C is observed, an action (also referred to as a treatment) Xt ∈ X is selected, and a response (called an outcome or reward in some situations) Yt ∈ Y is subsequently observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Z := C × X × Y is the (assumed time-invariant) sample space of the triple Zt := (Ct, Xt, Yt)—in this paper, we assume that Z is discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' see Remark 1 for an explanation of why as well as how to handle the case of continuous Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More formally, define the history at time t, Ht, to be the sequence ((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (Ct, Xt, Yt));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' H0 is simply defined to be empty ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then the action Xt is selected conditionally on Ht−1 and Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' These action- selection probabilities are encoded in the known decision-making (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', adaptive assignment) algorithm A, where the probability of selecting action x ∈ X at the tth timestep given the tth context and prior history 3 of context-action-response triples up until time t is given by PA(x|Ct, Ht−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The joint density of the full data sequence HT = ((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , YT )) is denoted by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Unless otherwise noted, we always consider Markovian systems, and thus assume that: Yt ⊥⊥ Ht−1 | (Ct, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 (1) In this paper, we also sometimes consider domains without contexts, in which case Ct = ∅ for all t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For a given data sequence (C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , YT ), set D := HT to be their concatenation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' take D := ZT to be the sample space of the random variable D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, as a note on general notation, we use [n] to refer to the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , n} and write [n : m] to denote {n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , m} for integers n ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Contribution In this paper, we provide exactly valid randomization-based inference for adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We make the following contributions, outlined below: We derive a weighted Monte Carlo (MC) randomization testing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This test is able to resample data from essentially any arbitrary resampling distribution (as long as a certain proportionality hypothesis is satisfied) and, by weighting the samples according to certain likelihood ratios, produce a p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Along the way, we show that we are also able to utilize an unweighted Markov Chain Monte Carlo (MCMC) randomization procedure developed in Besag and Clifford (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Barber and Janson (2022+), and applied in other settings, to perform inferential tasks on adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Both approaches offer exactly valid Type-I error control at essentially whatever computational cost the user desires (although, more computation generally results in higher power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Our novel weighted MC approach, however, is empirically no less powerful than the MCMC procedure and is, in fact, often more powerful in our simulations, especially when the number of resamples is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We use both the weighted MC and unweighted MCMC frameworks to test against a number of general null hypotheses on adaptively collected data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Letting g be any function of X, we can test Yt ⊥⊥ Xt | (Ct, g(Xt)) (2) in particular settings by resampling copies of Xt for each t ∈ [T] from any distribution conditional on g(Xt) and weighting these resamples by certain likelihood-based ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the special case in which g is a constant function, Xt can sometimes be sampled in such a way such that no weighting is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, the prior works of Pocock and Simon (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu (2022) are all able to handle this special setting by simply resampling the Xt’s by fixing the contexts and responses and re-running A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In general, however, when g is non-constant, this unweighted procedure cannot be applied, yet our procedure always can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One example of such a non- constant g might map treatments x in a multidimensional treatment space X to their first dimension x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The hypothesis being tested in equation (2) would then be of conditional independence of the response with all other dimensions of the treatment given the context and first treatment dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, our framework applies in challenging settings like MDPs, whereas that of prior work does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Beyond this added generality of our testing procedure, it offers various improvements (in terms of power and potential computational efficiency) over the existing work in certain settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We can test for non-stationarity in the conditional distributions Yt | Xt, Ct in various complex adaptive data collection settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under the semiparametric assumption that Yt | Xt, Ct follows a location model with location parameter additive in Xt, our first test can be inverted to construct confidence regions for the location differences 1This assumption will only become relevant in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, while this Markovianity assumption is standard in the adaptive data collection literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bibaut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021), we discuss in Remark 4 how our procedure can be applied to test slightly different hypotheses when no Markovianity assumption is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 4 between actions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Similarly, our non-stationarity test can be inverted to construct conformal prediction regions for YT under any adaptive assignment algorithm, considerably generalizing existing work which can only perform conformal inference under certain very specific adaptive assignment algorithms involving only a single round of adaptivity (Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Fannjiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2022) or can only do so approximately (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Gibbs and Cand`es, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Within our framework, it is more desirable (in terms of power) to have the weight of each resample as close as possible to that of the observed data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' developing resampling procedures which achieve this for the various inferential tasks and environments discussed above is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, we devise a number of novel and computationally tractable resampling procedures to be used for our tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, to demonstrate the practicality of our randomization tests (which can be deployed at essentially any user-specified computational load) as well as the statistical efficiency of our resampling procedures, we perform a simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' By considering the aforementioned inferential tasks in a variety of data- generating environments, and by using data collected by a number of decision-making algorithms—including both deterministic and randomized ones—we demonstrate that our resampling procedures produce a test which is both statistically efficient (in terms of power and average confidence/prediction interval length) and computationally tractable, while, of course, also guaranteeing exact validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 Related work 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Randomization tests for adaptively collected data As noted in Rosenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2019), the works of Pocock and Simon (1975) and Simon (1979) consider randomization testing for adaptively collected data consisting of actions and outcomes in the Markovian setting of equation (1) under a certain restricted set of assignment algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Extending their approach, Ham and Qiu (2022) are able to additionally handle context variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard (2019) also develop a randomization test that generalizes that of Pocock and Simon (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon (1979) primarily by relaxing the Markovianity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The key to the approach put forth in all of the aforementioned papers is that they restrict the adaptivity of the data collection environment and consider null hypotheses such that simply fixing the contexts and responses and resampling the treatments via the known adaptive assignment algorithm produces exactly exchangeable copies of the data, thereby enabling them to conduct a standard unweighted randomization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, our main result, the weighted MC randomization test (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1) makes no such assumptions about the data-generating distribution nor the resampling distri- bution other than that they satisfy a certain proportionality hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Also, as we empirically demonstrate in Section 5, our approach can be powerful even on data generated by deterministic assignment algorithms in the Markovian setting, while that of Pocock and Simon (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu (2022) are provably powerless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For more details regarding the comparison of our work to prior work see Remarks 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Inference in bandits and reinforcement learning Existing works for performing inference in rein- forcement learning settings are typically either asymptotic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Lai and Wei, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hadad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021, 2022) or, if not, are conservative in their finite- sample bounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Kaufmann and Koolen, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As opposed to these works, all our hypothesis tests are able to maintain exact and non-conservative validity in finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 MCMC and conditional permutation/randomization tests The connection between sampling exchangeable samples and MCMC was first developed by Besag and Clifford (1989) and has more recently been utilized by Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Barber and Janson (2022+), for the purposes of efficiently running a conditional permutation test, generating knockoffs, and approximate co-sufficient sampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The MCMC randomization test developed by Besag 5 and Clifford (1989) which we outline in Section 2 takes the same approach: it generates exchangeable samples using base draws from a Metropolis–Hastings Markov Chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, to the best of our knowledge, our work is the first that applies this MCMC approach to adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Conformal inference Our weighted MC randomization test, when applied to non-stationarity testing and inverted, has close connections with conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The works of Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020) and Fannjiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2022) are most related to our weighted MC randomization test when applied to non-stationarity testing and conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020) first extend conformal inference from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' data to the setting of covariate shift, under the assumption that the likelihood ratio between test and train covariates is known by developing a more general weighted conformal prediction (WCP) procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Relatedly, Hu and Lei (2020) apply WCP to give an asymptotically powerful non-stationarity test in the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The work of Fannjiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2022) extends Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020) and shows how to handle dependence between train and test sets in the setting of feedback covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' These prior methods handle only a single round of adaptivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', the first T − 1 rounds of “train-time” data is fully i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', but the “test-time” covariate distribution at timestep T may either be different (Hu and Lei, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020) or depend on the first T − 1 datapoints via feedback covariate shift (Fannjiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In contrast, our test for non-stationarity (the inversion of which yields a conformal prediction interval) is able to handle any number of rounds of arbitrary analyst adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Another related line of work is in handling fully non-exchangeable data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' that is, in contrast to Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020) and Fannjiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2022), no exchangeability is assumed within any particular train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2018), Gibbs and Cand`es (2021), Xu and Xie (2021), Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2022) all consider various versions of non-exchangeable data and extend conformal inference to these regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The methods presented in these works, however, are anti-conservative with bounds degrading as the degree of non-exchangeability grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In contrast, the methods presented in this paper (using the adaptivity known to the analyst) provide exact validity in the adaptive (and hence potentially highly non-exchangeable) settings in which they apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='5 Outline In Section 2 we introduce the weighted MC randomization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We also briefly describe how to perform an unweighted randomization test by using MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We then, in Section 3, show how our approach can be applied to solve a wide range of difficult inferential problems on adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then in Section 4 we introduce a number of novel algorithms for generating resamples that can be used for a variety of inferential tasks and adaptive data collection environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, in Section 5, we perform a simulation study of the methods introduced in Sections 2 and 3, using the resampling procedures from Section 4, that both empirically validates our methods by illustrating their validity and computational tractability, and demonstrates their power in a variety of challenging settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2 The weighted MC randomization test In this section we introduce our main approach to randomization testing: a novel weighted MC randomiza- tion test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, at the end of this section, we briefly review an unweighted MCMC randomization testing framework introduced by Besag and Clifford (1989), but, to the best of our knowledge, never before applied to adaptively collected data (as we do in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The key to both approaches is the ability to define likelihood-ratio-based weights that satisfy a certain proportionality hypothesis, which, as we show in Section 3, is possible in a range of inferential tasks in various adaptive data collection settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Empiri- cally, our weighted MC approach never performs worse than the unweighted MCMC test, and indeed often dominates it, especially when the number of resamples is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 6 Our weighted MC randomization test is a randomized procedure, taking the data set D as input, which sam- ples m conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' MC draws ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) given D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Define D to be the list ( ˜D(0), ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) where ˜D(0) := D and let q( ˜D(i)|D) denote the conditional probability of sampling ˜D(i) from this condi- tional distribution given that the dataset D was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Setting ˆf( ˜D(i)) := �T t=1 PA( ˜X(i) t | ˜C(i) t , ˜H(i) t ), the procedure then calculates likelihood-ratio-based weights for each resample ˜D(i) in D: wq D( ˜D(i)) = ˆf( ˜D(i)) � k∈[0:m]\\{i} q( ˜D(k)| ˜D(i)) �m j=0 ˆf( ˜D(j)) � k∈[0:m]\\{j} q( ˜D(k)| ˜D(j)) , (3) where ˜C(i) t , ˜X(i) t and ˜H(i) t are, respectively, the context at, action at, and history up until timestep t of the ith resample ˜D(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the procedure outputs the p-value p := m � i=0 wq D( ˜D(i))1[S( ˜D(i)) ≥ S(D)], (4) where S : D → R is any test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We summarize this procedure in the pseudocode of Algorithm 1 Algorithm 1: Weighted Monte Carlo randomization test Input: Data sequence D, resampling distribution ˜q, number of samples m, test statistic S 1 Sample ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) | D i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ∼ ˜q(·|D) 2 D ← ( ˜D(0), ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) where ˜D(0) := D 3 Compute wq D( ˜D(i)) for each i ∈ [0 : m] via equation (3) Output: p, calculated via equation (4) The above weighting scheme may not always yield a valid p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We show, however, that as long as the hypothesis H∝ 0 : ˆf( ˜D(i)) = Kf( ˜D(i)), ∀i ∈ [0 : m] for some constant K not depending on i, (5) holds, then p is a valid p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Importantly, note that the left hand side of the proportionality (5) is always computable (since it depends only on A, which is known), whereas the right hand side is not in general since the density f is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As we show in Section 3, H∝ 0 holds for a number of null hypotheses in various adaptive data collection settings, thereby allowing for the valid use of the above to test against such nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now present the main result of this paper, which is that, under H∝ 0 , p is a valid p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The p-value defined in equation (4) stochastically dominates the uniform distribution under equation (5) for any resampling distribution ˜q satisfying H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While we relegate the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 to Appendix A, we note here that the proof is quite different than those of randomization tests in the standard unweighted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For example, the proof of validity of the standard (unweighted) randomization test for exchangeable data and resamples—which is essentially an immediate consequence of the joint exchangeability of the original data and all resamples (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Besag and Clifford (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Edgington and Onghena (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Candes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021))—clearly will not apply in our case, since the data and resamples are not jointly exchangeable and the weighting scheme further makes it unclear how any such exchangeability could be used to prove validity of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A second, more recent proof technique, given by Hemerik and Goeman (2018) in the context of the MC permutation test on exchangeable data, is to condition on the set orb(D) of all possible permutations of the observed data and use the fact that, conditionally on orb(D), the original data is uniform over orb(D) and the set of resampled permutations is also uniform to show that the test is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In our setting, however, due to the non-exchangeability, this conditional distribution need not be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Rather, the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 is via the application of a simple yet new (to the best of our knowl- edge) application of Bayes’ theorem and proceeds by showing the conditional validity of p given the list D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Furthermore, as discussed in Remark 8, the assumption that the resamples ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) are drawn conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' given D can be relaxed and a more general test (with generalized weights) can be used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 7 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 given in Appendix A applies only to discrete data distributions f and resampling distributions q (recall we assumed all data was discrete in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We focus only on such distributions as in any real-world application of our test, the computer on which our test is being run on has only finite precision, rendering both the original dataset as well as all resamples discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, any practitioner running our test on a computer is necessarily dealing with discrete data, not due to any restrictions of the test itself, but rather the finite precision limitation of the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is not to say, however, that continuous distributions should never be considered in any situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For example, in an asymptotic theory, rates of convergence for discrete distributions may degrade as the resolution of discretization becomes finer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such a case, a continuous approximation of a computer-rendered discrete distribution may be considerably more useful than directly handling the discrete distribution itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The key difference in our case, however, is that our theory is exact in finite samples for any discrete distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, the guarantee of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 does not worsen at finer discrete resolutions, but rather remains intact, thereby permitting our consideration of only discrete distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note, however, that our procedure should apply much more broadly to continuous distributions, but perhaps not completely generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See (Huang and Janson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 5) for one such example in which our theory does not hold because Bayes’ theorem is violated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' by defining a dataset with T = 2 as well as a resampling distribution with m = 1 for which the marginal distribution of the resample P( ˜D(1)) is not equal to ˜q( ˜D(1)|D)f(D) by taking X1 ∼ Unif(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 1) and defining ˜q to sample ( ˜X(1) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(2) 1 ) ∼ LX1 where LX1 is the segment of length 1 orthogonal to the x-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' intersecting it at the point X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' with an angle of (1 − X1)10π/2 with the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 2 (Randomizing for exact Type-I error control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Just as with a standard randomization test and, as a special case, conformal inference in which the procedure is called smoothing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2005), one can define a “lower” p-value p− := m � i=0 wq D( ˜D(i))1[S( ˜D(i)) > S(D)] and randomize to obtain a test for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 which controls Type-I error at exactly the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, the test which fails to reject when p− > α, rejects with probability α−p− p−p− when p− ≤ α < p, and otherwise always rejects, controls Type-I error at exactly the nominal level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 gives one great freedom and generality in computing valid randomization p-values by allowing for many choices of the conditional distribution q, two aspects in particular remain, at present, quite unclear: (a) in what situations and under what null hypotheses can we ensure that the proportionality hypothesis H∝ 0 holds, as well as (b) which choices of q one should sample from to obtain a powerful test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We address the first question in the context of adaptively collected data in Section 3 and discuss the second in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The unweighted MCMC randomization test Above, we discussed a weighted MC randomization test which weights resamples based on certain likelihood ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Here, we briefly outline an unweighted MCMC randomization test developed by Besag and Clifford (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While the test itself is not new (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Barber and Janson (2022+) for more recent utilizations and extensions of the test), our application of it to adaptively collected data, to the best of our knowledge, is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The randomized procedure for the unweighted MCMC randomization test repeatedly takes Metropolis–Hastings steps, starting at D by, at each step i, sampling a proposal ˜D given ˜D(i−1) from q(·| ˜D(i−1)), and additionally computing an acceptance ratio under the stationary distribution f to decide if the Metropolis–Hastings step accepts the proposal or remains at ˜D(i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Just as with Algorithm 1, the acceptance ratio calculation actually only involves known quantities depending on ˆf and q and hence the validity of the MCMC test is again implied by the proportionality hypothesis H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Besag and Clifford (1989) for a delineation of the general test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We also note here that, in all of our simulations, our weighted MC test performs no worse than—and often dominates—the unweighted MCMC test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, exploring the utility of our weighted MC test in the settings of Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Barber and Janson (2022+), all of which use the unweighted MCMC procedure, may be of interest for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 8 3 Randomization testing for adaptively collected data In this section, we apply the randomization tests from Section 2 to solve a host of challenging inferential tasks in adaptive data collection settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Section 2, we saw that to perform a weighted MC randomization test we needed to be able to run Algorithm 1 which in turn required us to choose a q that satisfies the hypothesis H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this section, we show that, for a number of null hypotheses in various adaptive data collection settings, H∝ 0 can be satisfied with many non-trivial choices of q, thus allowing for the application of the weighted MC randomization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that, since the focus of this section is ensuring that H∝ 0 holds (under various null hypotheses and adaptive data collection settings), all results also immediately apply to the unweighted MCMC test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We address two broad classes of tasks and describe each—as well as the adaptive data collection environments in which we consider them—in detail in the sections that follow: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Testing conditional independence between treatment and response conditional on context and some function of the treatment (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Applying certain transformations to the data and inverting such tests allows us to construct confidence intervals for response distribution parameters in certain semiparametric models (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Non-stationarity testing (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2) and its application to conformal inference (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2) In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 below, we discuss a number of adaptive data collection environment setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Each of these setups assumes some combination/subset of equations (1) (Markovianity), (6), (7a), (7b): Yt | Ct, Xt does not depend on t (Y -Stationarity) (6) This first assumption asserts that the environment is Y -stationary so that the conditional response distribu- tion does not vary with time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' this distinction will be important in whether or not additional randomization can be employed for a more powerful conditional independence test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ct ⊥⊥ (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Xt−1) | (C1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ct−1, Yt−1) (Weak non-reactivity) (7a) Ct | Ht−1 depends neither on t nor Ht−1 (C-stationarity & strong non-reactivity) (7b) The assumptions of equations (7a) and (7b) are related and describe the environment through the behavior of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, the weak non-reactivity assumption (equation (7a)) says that the actions prior to time t do not affect Ct, conditionally on the previous contexts and responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', the future states of the environment are essentially (conditionally) non-reactive to prior actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This weak non-reactivity assumption is also made in (Ham and Qiu, 2022, Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The C-stationarity & strong non-reactivity assumption (equation (7b)) is a stricter version of this, and stipulates that each context is generated by the environment independently from the past and also that the distribution from which it is generated does not change with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Before proceeding, we pause here to emphasize that all results that ensue in this section hold for any adaptive assignment algorithm A, as long as it is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We present all results by first describing the null hypothesis being tested, then explaining the various environments in which it can be tested, and finally delineating any constraints on the resampling distribution q in each of these environments so as to ensure that H∝ 0 holds under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Conditional independence testing Null hypothesis Let g be any function of x ∈ X, with unspecified codomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We wish to perform a conditional independence test between treatment and response given both context and the g-evaluation of 9 the treatment: H⊥⊥,g 0 : Yt ⊥⊥ Xt | (Ct, g(Xt)), ∀t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One example in which this hypothesis might be employed is in the problem of A/B testing in online advertis- ing, where a company presents users with the same ad but with differing hues, saturations, and brightnesses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', so the treatment space is 3-dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One concrete hypothesis that may be tested in this situation is if saturation and brightness are enough to predict user response alone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', is user response independent of hue given both saturation and brightness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Setting g(Xt) = (Xt,saturation, Xt,brightness) to map Xt to its saturation and brightness components, H⊥⊥,g 0 is equivalent to this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A second scenario is in testing if a particular subset of treatments induces the same response distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, letting X = {0, 1, 2} be the treatment space, the null hypothesis that Yt ⊥⊥ Xt | (Ct, Xt ∈ {0, 1}) is equivalent to H⊥⊥,g 0 with g(Xt) = I(Xt = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, in the special case in which g is a constant function, the hypothesis being tested is that of simple conditional independence between treatment and response given context, and, for this par- ticular choice of g, the unweighted test of prior work (Pocock and Simon, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu, 2022) can be employed in Environments 1 and 3 (but not in Environments 2 or 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' as we describe below, however, our framework allows for a significant improvement in power over this prior work when employed in Environment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Environment 1 (Non-reactive environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The non-reactive environment is one in which the Marko- vianity (equation (1)) and weak non-reactivity (equation (7a)) assumptions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Examples of a non-reactive environment include (contextual) bandits—a setting that is common in the mobile health literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Tewari and Murphy, 2017)—as well as many common adaptive experimental designs, such as adaptive crop yield experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Alesso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Environment 2 (Markov Decision Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The next environment we consider is an MDP, which assumes only the Markovianity assumption (equation (1)) holds, and also stipulates that Yt := Ct+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' note that this implies that Assumption (7a) also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One example of MDP data arises in electronic health records (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Environments 3 and 4 below are Y -stationary counterparts of Environments 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Environment 3 (Stationary non-reactive environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The most restricted of the environments we con- sider, the stationary non-reactive environment, assumes Markovianity (equation (1)), Y -stationarity (equa- tion (6)), and C-stationarity & strong non-reactivity (equation (7b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Examples of this environment arise in reinforcement learning as a stationary (contextual) bandit as well as in many common adaptive experimental designs—one example being in adaptive clinical trials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Environment 4 (Stationary MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' With the additional assumption of stationarity, the stationary MDP is a special type of MDP in which Y -stationarity (equation (6)) also holds, thereby assuming that the MDP’s transition distribution does not change across timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' An example of stationary MDP data is in patient admissions to the emergency room during surge demand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Lee and Lee, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 A second example arises in robotics, where the robot’s dynamics and interactions with its environment can be modeled as a stationary MDP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Su´arez-Hern´andez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Constraints on q The resampling procedure q may incorporate two sources of randomization: (a) ran- domizing the order of the data by some permutation in a certain environment-specific set Π, and (b) ran- domizing the actions conditional on their g-evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More specifically, q must be such that the sequences of contexts, responses, and g-evaluations of the treatment in each resample are equal to some (random) 2To distinguish this setting from the adaptive crop yield experiment example, note that in an adaptive clinical trial, patients are selected i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from some population, thus obeying the C-stationarity & strong non-reactivity (equation (7b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, in an adaptive crop yield experiment, due to weather fluctuations (such as temperature and precipitation) over time, this assumption is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For the same reason, we expect adaptive clinical trial settings, but not adaptive crop yield experiments, to be Y -stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3Again, this example differs from that of electronic health records in Environment 2 because here, we do not expect the transition dynamics to change over time, whereas in the previous example, administering certain medication to a patient may in fact change their internal state and thus alter the way in which they react to the same medication later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 10 permutation—albeit, the same one—π ∈ Π of their respective sequences in the original data: � ˜C(i) t , g( ˜X(i) t ), ˜Y (i) t � = � Cπi(t), g(Xπi(t)), Yπi(t) � , ∀t ∈ [T], for some πi ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (8) Note that equation (8) does not force ˜X(i) t = Xπi(t), and thus allows the treatments to be further randomized over X, so long as the restriction g(Xπi(t)) = g( ˜X(i) t ) is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the restricted set Π of allowable permutations is environment-specific;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we first focus on Environments 1–3, where the non-stationarity of Environments 1 and 2 prevent us from permuting the data at all, while the stationarity of Environment 3 allows for any permutation: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Let Πi, i ∈ [3] denote the set of allowable permutations for Environments 1-3, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then if Π1 = Π2 = {id}, where id denotes the identity permutation, and Π3 = Π[T ], the set of all permutations on [T], the proportionality hypothesis H∝ 0 is satisfied under H⊥⊥,g 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Environments 1 and 2, the restriction that Π1 = Π2 = {id} ensures that ˆf( ˜D(i)) = T � t=1 PA( ˜X(i) t | ˜C(i) t , ˜H(i) t−1) = �T t=1 Pt(Yt|Ct, g(Xt), ˜X(i) t )PA( ˜X(i) t | ˜C(i) t , ˜H(i) t−1)P(Ct|C1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ct−1, Yt−1) �T t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ct−1, Yt−1) by H⊥⊥,g 0 and that g( ˜X(i) t ) = g(Xt) = �T t=1 Pt( ˜Y (i) t | ˜C(i) t , ˜X(i) t )PA( ˜X(i) t | ˜C(i) t , ˜H(i) t−1)P( ˜C(i) t | ˜C(i) 1 , ˜Y (i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜C(i) t−1, ˜Y (i) t−1) �T t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ct−1, Yt−1) as Π1 = Π2 = {id} & equation (8) = 1 �T t=1 Pt(Yt|Ct, g(Xt), Xt)P(Ct|C1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ct−1, Yt−1) f( ˜D(i)), due to equation (7a) where the subscript t on P is used to emphasize that the conditional distribution Yt | Ct, Xt may depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, the proportionality hypothesis is satisfied with K = 1 �T t=1 Pt(Yt|Ct,g(Xt),Xt)P(Ct|C1,Y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',Ct−1,Yt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' in Environment 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Π3 = Π[T ] and any permutation is allowed since ˆf( ˜D(i)) = T � t=1 PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) = �T t=1 P(Yπi(t)|Cπi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xπi(t))PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1)P(Cπi(t)) �T t=1 P(Yt|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt)P(Ct) due to equations (6) and (7b) = �T t=1 P(Yπi(t)|Cπi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) t )PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1)P(Cπi(t)) �T t=1 P(Yt|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt)P(Ct) by H⊥⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g 0 and that g( ˜X(i) t ) = g(Xπi(t)) = �T t=1 P( ˜Y (i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) t )PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1)P( ˜C(i) t ) �T t=1 P(Yt|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt)P(Ct) by equation (8) = 1 �T t=1 P(Yt|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt)P(Ct) f( ˜D(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' due to equation (7b) where we have dropped the subscript t on P used above (and also do not need to subscript P(Cπi(t)) due to the Y -stationarity assumption of equation (6) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the C-stationarity assumption of equation (7b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, once again, the proportionality hypothesis holds, but now with K = 1 �T t=1 P(Yt|Ct,Xt)P(Ct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The precise definition of the allowable set of permutations Π4 in Environment 4 is somewhat more complicated due to the serial dependence between consecutive timesteps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Ct+1 is generated conditionally on (Ct, Xt)) 11 that is not present in Environment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, only permutations which preserve the property that the tth response is equal to the (t + 1)th context are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We formally state a Proposition here, but relegate its proof to Appendix B: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Setting Π4 := {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1}, the proportionality hypothesis H∝ 0 is satisfied under H⊥⊥,g 0 in Environment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We briefly note here that unless the MDP’s state space C is relatively small, then the (data-dependent) permutation set Π4 will typically be quite small and virtually no randomization in permutations can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' There are, however, settings in which the state space C is small, such as in the restless multi- armed bandit considered by Mate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2020, 2021) in the context of patient adherence monitoring and well-being, in which our test can be used effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 3 (Comparison with existing work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Environment 1 (and Environment 3, which is a special case), when g is a constant function, the unweighted randomization test of Pocock and Simon (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu (2022) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, the testing procedure presented above has two advantages over those in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' First, by randomizing timesteps in addition to treat- ments, our procedure is more powerful than theirs in the stationary setting of Environment 3, especially for deterministic assignment algorithms for which their procedure is powerless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we empirically demonstrate this in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Second, their procedure requires that each resample rerun A independently m times with the same sequence of Ct and Yt as in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 While this would seem to be the most natural and statistically powerful approach, it may not always be computationally feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such a case, our method, by using a more easily computable resampling procedure, provides a computationally tractable workaround (albeit perhaps at the cost of degraded statistical efficiency per resample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As a concrete example, the adaptive assignment algorithm A used to generate the original data could be based on Thompson sampling in a complex Bayesian model, thus rendering it too computationally burdensome to run for any but a very small number of MC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, unnormalized densities—which are all that our procedure requires, due to the proportionality assumption H∝ 0 —are generally easy to compute, thus rendering computation of the weights in equation (3) tractable under a less computationally intensive resampling procedure q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 4 (Relaxing structural assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Appendix C, we show the above testing procedure can be generalized to arbitrary adaptive data collection processes in which none of the assumptions of equations (6),(7a),(7b) nor the Markovianity assumption of equation (1) are made, and the adaptive data collection environment is assumed to be completely general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such an environment, we show that one can, somewhat analogously, consider a sequence of functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , gT and test simultaneous (over t) conditional indepen- dence between the tth context (as well as the tth response) and the prior sequence of actions given the prior sequences of contexts and responses as well as the sequence comprising the gs-evaluation of the sth action for s ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The special case of this hypothesis wherein all the gt are constant allows for unweighted random- ization testing in the completely general environment described in this Remark as shown by the prior work of Bojinov and Shephard (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Testing for non-stationarity Null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For our non-stationarity test, the null hypothesis HS 0 is that the response distribution is stationary (but unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, the conditional distribution of response given the context and treatment is the same across all timesteps: HS 0 : Yt | (Ct, Xt) does not depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this section, we consider two environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The first is a a special type of non-reactive environment (Environment 1): 4Pocock and Simon (1975) and Simon (1979) only describe how to do so for certain adaptive assignment algorithms, and both, as well as Bojinov and Shephard (2019), only consider the non-contextual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 12 Environment 5 (C-stationary strongly non-reactive environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The C-stationary strongly non-reactive environment is an environment in which equation (7b) holds, in addition to the Markovianity assumption of equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Once again, (contextual) bandits and various adaptive experimental designs are special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One concrete example is in adaptive experimental designs studying the effects of job search assistance programs on helping job seekers find work (Caria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020) over short periods of time since, in these brief time intervals, we expect the “context” surrounding each individual (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', background, credentials, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=') to be roughly i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One additional example of a C-stationary strongly non-reactive environment for which our theory also holds is an episodic MDP, in which each episode is viewed as a single time step, the reward sequence a single response, and the action sequence a single (high-dimensional) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='5 The second environment we consider is simply the MDP of Environment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Constraints on q By a proof similar to that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 for Environment 3, it is straightforward to see that, as long as each draw from the resampling distribution q is (any) permutation of the original data D, the proportionality hypothesis H∝ 0 holds under HS 0 in Environment 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, as long as we set Π = Π[T ] and do not allow for any other randomization, then ˆf( ˜D(i)) ∝ f( ˜D(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under the MDP setting of Environment 2, we may use the same randomization set of permutations Π4 stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 and again, do not include any additional randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We summarize this below: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' If Π = {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1} in Environment 2 and Π = Π[T ] in Environment 5, and the only randomization in q’s resampling consists solely of permutations drawn from Π, then the proportionality hypothesis H∝ 0 is satisfied under HS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Inverting tests to construct confidence and prediction intervals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Confidence regions in semiparametric models Recall that the hypothesis tests in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 were all nonparametric and focused on testing (conditional) independence and distributional equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this section, we turn to the problem of exact parametric inference in semiparametric models—focusing on semiparametric location models as a case study—and use a technique previously used in standard randomization testing to construct confidence intervals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Rabideau and Wang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Before proceeding, we emphasize that parametric inference for adaptively collected data is a challenging problem and, as far as we are aware, all examples in the literature are either asymptotic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hadad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021), or conservative (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Kaufmann and Koolen (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Now, consider the setting in which the response distribution is distributed according to a location family with locations determined by action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More precisely, letting h0 denote a (unknown) base density, we assume that at each time-step t ∈ [T], Yt | (Xt = x) ∼ h0(y − θx), where x �→ θx is some mapping of actions to location parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such a setting it is quite natural to ask: how much better or worse is action x compared to x′, in terms of their location parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More formally, how can we test against the null hypothesis HLoc,δ,x,x′ 0 that θx′ − θx = δ for actions x, x′ ∈ X and δ ∈ R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To test against HLoc,δ,x,x′ 0 , we modify the dataset D and then perform a conditional independence test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, by modifying the dataset by replacing Yt with Yt + δ · 1[Xt = x], we get that HLoc,δ,x,x′ 0 implies that x and x′ induce the same distribution over rewards in the modified data generating distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence a test against H⊥⊥,g with g(Xt) = � {x, x′} if Xt ∈ {x, x′} Xt otherwise on this modified dataset serves as a test against HLoc,δ,x,x′ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' These same ideas can also be applied to scale families;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' see Appendix D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Most importantly, these tests can all be inverted to construct confidence regions for the parameter in question, namely θx′ − θx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For example, consider constructing a confidence region for the difference in locations of 5This episodic MDP setting also falls under the category of a (stationary) non-reactive environment, and hence our tests of conditional independence presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 also apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 13 treatments x and x′ at nominal miscoverage rate α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One can construct the acceptance region by simply running the test against HLoc,δ,x,x′ 0 at level α at each δ in the parameter space ∆: those δ for which the test fails to reject make up the acceptance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In cases where the parameter space ∆ is either continuous or too large to iterate over completely, approximate confidence regions can be constructed as follows: (a) grid the space into a small finite set of discrete points ∆′ ⊆ ∆, (b) run the above procedure at each δ ∈ ∆′ to obtain an accepted set of grid points A′, and (c) include all δ ∈ ∆ within a certain user-specified distance of any of the accepted grid points of A′ in the confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Prediction regions for YT Similar to the previous section, the tests of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 can be applied to construct prediction intervals for YT before it is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, again, one can construct the acceptance region for YT by running the non- stationarity tests described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 on the almost-fully realized dataset ((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , y)), with y ranging over Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' just as with confidence intervals, in the case of large Y, the gridding/rounding procedure described at the end of the previous section can be used to construct approximate prediction intervals by simply selecting a small discrete set Y′ ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We do note, however, that it is not uncommon for Y to be small or even categorical in many (challenging) settings and thus, for such problems, we can grid Y directly to obtain exactly valid prediction regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now make a few remarks about some broader connections of our procedure when used in this fashion to conformal inference, as well as challenges and speedups in constructing intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 5 (Conditional conformal inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While conditional conformal inference is impossible in general, even in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' data (Lei and Wasserman, 2014), Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 admits a conditional version which may be useful when the covariate space X is finite and small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, the validity of the test still holds when we replace f with the conditional density f(·|XT ), which conditions on the “test-time” covariate XT = xT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' thus, when inverting the hypothesis test and constructing a conformal prediction interval, one can guarantee valid coverage conditional on XT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', P(YT ∈ Cα T (XT )|XT ) ≥ 1 − α, where Cα m(XT ) denotes conformal band at XT with nominal miscoverage rate α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In practice, this is possible as long as each resample ˜D(i) has ˜X(i) T = XT (in addition to ˜D(i) being an allowable permutation of D), so that f( ˜D(i)|XT ) � k∈[0:m]\\{i} q( ˜D(k)| ˜D(i)) �m j=0 f( ˜D(j)|XT ) � k∈[0:m]\\{i} q( ˜D(k)| ˜D(j)) = f( ˜D(i)) � k∈[0:m]\\{i} q( ˜D(k)| ˜D(i)) �m j=0 f( ˜D(j)) � k∈[0:m]\\{i} q( ˜D(k)| ˜D(j)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In turn, p’s validity once again ensues so long as the proportionality hypothesis H∝ 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 6 (Challenges with explicit construction of conformal bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Split conformal inference (Pa- padopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2007) is a variant of conformal inference which involves data splitting in order to construct a conformal prediction region which can be explicitly and efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Unfortunately, such an ex- plicit construction of a conformal band using data splitting does not carry over using our procedure to test non-stationarity, since the weights, through ˆf( ˜D(i)), may depend on the test-time response grid values y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Sim- ilarly, the discretization procedure of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (2018) used to construct explicit bands in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' setting by rounding the response to a small discrete set is not valid in our framework since, in general, probabilities under PA involving the rounded data are either unknown or not efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 7 (Sharing samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To address the challenges of Remark 6, one can grid the space into a small finite set Y′ and run the test at each y ∈ Y′, as discussed above in the case of continuous or large Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Naively, however, this involves drawing m resamples for each y ∈ Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To reduce the total number of resamples needed, we can however share resamples between different values of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, for y1, y2 ∈ Y′, resamples drawn in association with y1 can be used to determine whether or not to accept y2 and vice versa, thereby more effectively using each resample drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 14 4 Resampling procedures In Section 3, we focused on an information-theoretic question: in what data regimes can we define a sampling procedure q for which our testing procedure is valid?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Here, we turn to the second key question posed at the end of Section 2: which choices of q are statistically best?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, while the last section specified how to choose the the support of q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', which variables should be randomized over and which should be held fixed) depending on the null hypothesis being tested against and the adaptive data collection environment in which it is being tested, this section explores what distributions to choose over these supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the remainder of this section, we provide a partial answer to this challenging question by proposing a number of resampling procedures that are compatible with the constraints outlined in Section 3 and will be shown to be powerful and produce short confidence/prediction regions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Recall that we are focusing solely on conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling from procedures q and thus all re- sampling/proposal distributions considered in this section can be applied to both the weighted MC and unweighted MCMC tests (with q as the proposal distribution for the MCMC test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, as mentioned in Section 2, our weighted MC procedure does not in general require conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling and hence the following remark describing how our test can be generalized in this case is in order: Remark 8 (Non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' When the resamples ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) are not generated in a conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' manner, the test in Algorithm 1 can be slightly generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, letting Σ be any subset of Π[0:m], the set of permutations on [0 : m], and ˜q(( ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m))|D) denote the conditional probability of sampling ( ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) given that D was observed, the procedure can be generalized by redefining the weights to be w˜q,Σ D ( ˜D(i)) = ˆf( ˜D(i)) � π∈Σ:π(0)=i ˜q(( ˜D(π(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(π(m)))| ˜D(i)) �m j=0 ˆf( ˜D(j)) � π′∈Σ:π′(0)=j ˜q(( ˜D(π′(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(π′(m)))| ˜D(j)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We relegate a description of this generalized procedure, as well as the proof of its validity, to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 9 (Computational issues with conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We also note here that even with a conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling scheme q, the computation of the p-value p can sometimes take Ω(m2) computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is because, even with a conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling procedure, all pairs of conditional densities q( ˜D(i)| ˜D(j)) must be computed and incorporated into the p-value computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, if the conditional density draws samples ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) such that q(·| ˜D(i)) is the same for all i ∈ [0 : m], then � k∈[0:m]\\{i} q( ˜D(k)| ˜D(i)) = �m k=0 q( ˜D(k)| ˜D(i)) q( ˜D(i)| ˜D(i)) ∝ (q( ˜D(i)| ˜D(i)))−1 (9) and so the p-value can be computed much more quickly in only O(m) computations by replacing � k∈[0:m]\\{i} q( ˜D(k)| ˜D(i)) with (q( ˜D(i)| ˜D(i)))−1 in the weight equation (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' as a result, we focus on resampling procedures that have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The setup in this section is that we first consider resampling procedures used to test non-stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We then go on to discuss resampling algorithms for testing the conditional independence null hypothesis H⊥⊥,g 0 , some of which use some of the non-stationarity testing resampling algorithms as subprocedures in their sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' All resampling procedures presented in this section are evaluated in our simulations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, more detailed pseudocode outlines of the resampling procedures in this section can be found in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Lastly, we make a brief note about the computation of probabilities under the resampling distribution q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' All resampling procedures we consider in this section—and in our simulations in Section 5— either sample uniformly or involve some sort of sequential sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In either case, computation of conditional probabilities under q are tractable as they are constant in the former and, in the latter, can be calculated as the sample is generated by serially multiplying together the corresponding conditional probabil- ities of each sequential sample as it is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Of course, probabilities of the form q(D| ˜D(i)) for i ∈ [m] 15 still must be computed (as the original dataset D is never resampled from q) from scratch, but this is simply done in the same manner as above: behaving as though D had indeed been sampled from q conditionally on ˜D(i) and sequentially multiplying conditional probabilities of each resampleed timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Non-stationarity testing in a C-stationary strongly non-reactive environ- ment We first describe four types of resampling procedures that can be used for non-stationarity testing in a C-stationary strongly non-reactive environment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Environment 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, all such distributions must only randomize timesteps by permuting the data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Apart from uniform permu- tations, the other three procedures discussed in this section randomly permute the data in a way intended to mimic A while also ensuring diverse random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We thus call these three resampling procedures imitationπ, re-imitationπ, and cond-imitationπ, the common word imitation referencing the mimicking of A (the prefixes re and cond will be explained later on in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' All three procedures randomly permute the data by serially sampling not-yet-sampled timesteps from the original data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' uniformπ sampling The uniformπ sampling procedure simply selects a permutation of the data uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Although simple, intuitively it may result in a diverse set of resamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' imitationπ sampling This sampling procedure samples permutations by sequentially resampling timesteps from the original data, where the sampling distribution at time t acts as though the first t − 1 already- resampled timesteps were drawn according to the true data generating-process and selects the tth timestep proportionally to the probabilities dictated by PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In other words, at each timestep t, letting R denote the set of not-yet-sampled timesteps, the imitationπ distribution draws a timestep t′ from R, where the proba- bility of drawing any given s ∈ R is proportional to PA(Xs|Cs, ˜Ht−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' if PA(Xs|Cs, ˜Ht−1) = 0 for all s ∈ R, the procedure is ended and an attempt at a new resample can be begun using the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, (Ct′, Xt′, Yt′) is appended to ˜Ht−1 and t′ is removed from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Intuitively, the imitationπ distribution behaves as A would, feeding in the already-realized sequence of responses (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , YT ), except that it may only sample amongst actions which correspond to not-yet-selected timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 3 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The re-imitationπ and cond-imitationπ distributions which we describe below are intended only for random- ized decision-making algorithms A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' When applied to a deterministic algorithm, they are both the same as imitationπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' re-imitationπ sampling This distribution is similar to the imitationπ distribution, except that, to incor- porate more diversity, it independently regenerates the exogenous randomness that the randomized decision- making algorithm A makes as it mimics it to draw resamples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' it thus rerandomizes the randomness of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More specifically, the re-imitationπ distribution views A as a sequence of decision rules δt which take as input the tuple (Ct, Ht−1, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ut), where Ut is the exogenous random variable generated by A at time t, and output which action to take: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Sample a permutation of D by sequentially resampling timesteps from the data, where the sampling distribution at time t uses the t − 1 already-resampled timesteps as well as the resampled exogenous randomness ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut−1, and then generates the random variable ˜Ut from Ut’s distribution, but con- ditional on the remaining timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, ˜Ut is sampled from the conditional distribution of Ut given that Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut−1, Ut) for at least one not-yet-selected timestep s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' If, however, this conditional distribution is degenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', because there does not exist ˜Ut for which Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut−1, ˜Ut) for any remaining timesteps s), the process is terminated and sampling for a new resample is begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Select the tth timestep uniformly over all those not-yet-sampled triples (Cs, Xs, Ys) with Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (10) 16 The motivation behind the re-imitationπ distribution, as opposed to imitationπ, is that, by incorporating more of the randomness used in the decision-making algorithm one may be able to obtain more diverse samples while also better imitating the decision-making mechanisms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 4 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' cond-imitationπ sampling The cond-imitationπ distribution is the same as re-imitationπ except that instead of resampling the ˜Ut’s, it conditions on them and thus uses the same sequence of exogenous random- ness as re-imitationπ does;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' this, of course, requires knowing the original sequence U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Intuitively, this conditioning that cond-imitationπ sampling performs should bias the weights closer to (m + 1)−1, resulting in more powerful resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 5 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Non-stationarity testing in an MDP We finally briefly discuss the resampling procedures for non-stationarity testing in an MDP (Environment 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We consider essentially the same four types of resampling procedures used for non-stationarity testing in a C-stationary strongly non-reactive environment described in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The only difference however, is that, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, only a subset of permutations are allowed in the MDP setting so as to ensure that the ˜Y (i) t = ˜C(i) t+1 condition that holds for i = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', in the observed data) also holds for each resample ˜D(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, while the four procedures which we consider here—also called uniformπ, imitationπ, re-imitationπ, and cond-imitationπ—sample timesteps serially without replacement according to A as their analogs did in the previous section, they do so over only a restricted subset of not-already-sampled timesteps at each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 For this reason, the MDP data that we consider includes an additional action XT +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' hence the permutations which we consider can be viewed as permuting the T +1 state-action pairs in this augmented dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='7 More specifically, suppose that at the end of round t − 1, the state-action pair (Cs′, Xs′) had just been selected and appended to ˜Ht−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then, the set of allowable timesteps which can be sampled at round t are only those not-yet-sampled state-action pairs (Cs, Xs) for which the state-action-next state triple (Cs′, Xs′, Cs) is present in the original data sequence D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The precise way in which the timestep is sampled is then in exact accordance with the weighting, randomization, and conditioning that correspond to the uniformπ, imitationπ, re-imitationπ, and cond-imitationπ sampling procedures described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithms 6, 7, 8, and 9 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Conditional independence testing We now describe four types of resampling procedures that can be used in a stationary non-reactive environ- ment (Environment 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Three of these resampling procedures randomize both timesteps and the action Xt conditional on g(Xt) allowed by the stationarity, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The procedure which does not both randomize timesteps and actions simply randomizes the actions alone and is called imitationX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' as such, it can also be applied to the non-reactive environment (Environment 1) as well as an MDP (Environment 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Of the three procedures which randomize both timesteps and actions, two randomize the timesteps and actions in two separate stages and therefore use some of the resampling procedures discussed in the previous section (as well as one more) in the first stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we call these resampling procedures uniformπ+imitationX and restricted-uniformπ+imitationX (the latter of these two uses a permutation scheme that involves g and was thus not discussed in the previous two sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' These resampling procedures can all be applied to a station- ary MDP (Environment 4), by using the analogous MDP permutation distribution as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that we do not consider the combinations imitationπ+imitationX, re-imitationπ+imitationX, and cond-imitationπ+imitationX because, while conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', they violate the property discussed in 6For uniformπ permutations, we sequentially sample indices uniformly at random from these restricted subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 7Note that, practically speaking, if the dataset D in consideration does not have this additional action XT +1, then the analyst can add it with ease (because they know the adaptive assignment algorithm A) and can do so without affecting the null or alternative distribution from which the data was drawn (because the null/alternative distributions govern only the transition dynamics, and not action selection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 17 Remark 9 that q(·| ˜D(i)) is the same for all i ∈ [0 : m], and hence require Ω(m2) computations render- ing them somewhat computationally burdensome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The fourth resampling scheme combines the two stages of permuting timesteps and randomizing Xt into a single procedure and is thus referred to as combinedπ,X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' this procedure applies in the stationary non-reactive environment (Environment 3) and can be modified to work in a stationary MDP (Environment 4) by using the usual sequential permutation restriction described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, just as in the previous two sections, all of these distributions are based on the idea of trying to draw resamples in a way that mimics the behavior of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' imitationX sampling The imitationX distribution, at each timestep t, conditions on the t − 1 already- resampled data points ((C1, ˜X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (Ct−1, ˜Xt−1, Yt−1)) as well as Ct and, treating them as though they were true data points sampled by A, samples ˜Xt amongst all x ∈ X with g(x) = g(Xt), weighting proportionally to the action-selection probabilities of PA—if all weights are 0, then the process is exited and an attempt at a new resample can be begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The intuition behind this sampling procedure is that it attempts to mimic A by essentially feeding in the already-realized context, g-evaluation, and response sequences to generate the sequence of actions (each sampled conditionally on the g-evaluation at the corresponding timestep), thereby mimicking the true data-generating distribution induced by A (resulting in weights closer to (m + 1)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that the imitationX resampling algorithm when applied in Environments 1 and 3 yields precisely the same (unweighted) test as the prior work of Pocock and Simon (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 11 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' uniformπ+imitationX sampling This resampling procedure proceeds in two stages: the first stage ap- plies a uniform permutation using the uniformπ sampling of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 and the second randomizes the treatment conditional on its g-evaluation in the permuted data sequence using the imitationX resampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Similar to imitationX, we intuitively expect such a sampling procedure to have weights near (m + 1)−1 while also incorporating significant diversity (resulting in more varied evaluated test statistics S( ˜D(i))) due to the initial uniform permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 12 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' restricted-uniformπ+imitationX sampling This procedure is identical to uniformπ+imitationX sam- pling, except that it modifies the uniform sampling in step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, for this resampling procedure, the randomly sampled permutation in the first stage is a restricted uniform permutation, which does not permute the sequence of g-evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In other words, only permutations π for which g(Xt) = g(Xπ(t)) for all t ∈ [T] are allowed, and the sampling is uniform over this restricted set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The intuition behind this sampling scheme is similar to that of the uniformπ+imitationX sampling procedure except that, by using restricted uniform permutations rather than fully uniform permutations, the sampled data appears more similar to the original data with the goal of making the weights closer to (m + 1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 13 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' combinedπ,X sampling As opposed to the previous two sampling schemes which permute timesteps and randomize treatments conditional on their g-evaluations in two separate stages, this sampling approach combines both types of randomization into a single resampling stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, at each timestep t, the combinedπ,X resampling procedure samples t′ according to the imitationπ distribution from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 on g- evaluations over not-yet-selected timesteps, again, by conditioning on the already-resampled data, and then randomizes Xt′ conditional on g(Xt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In other words, at timestep t, the combinedπ,X resampling proceeds by: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Selecting a not-already-selected timestep t′ conditionally on ˜Ht−1 via the imitationπ distribution of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 applied to g-evaluations of actions (instead of simply the actions themselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' If PA(g(Xs)| ˜Ht−1, Cs) = 08 for all not-yet-selected timesteps s, then no such sample t′ can be gener- ated and so the sampling process is terminated and a new one may be begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Once the timestep (Ct′, Xt′, Yt′) is selected, ˜Xt′ is sampled via PA(·| ˜Ht−1, Ct′, g(Xt′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 8By PA(g(Xs)| ˜Ht−1, Cs) we mean the probability (induced by PA) that the g-evaluation at the tth timestep is equal to g(Xs) given the history ˜Ht−1 and context Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 18 Similar to restricted-uniformπ+imitationX sampling, the intuition behind combinedπ,X sampling is that both randomization across timesteps and the treatments, conditional on their g-evaluations, are incorporated, except that here they are combined and both the timestep and first action component randomization mimic A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' See Algorithm 14 for pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5 Empirical Results In Section 4 we proposed a number of resampling procedures q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this section, we implement these resampling procedures in a variety of adaptive data collection environments—which are all special cases of the more general environments discussed in Section 3 for which our theory holds—and both demonstrate their validity and evaluate their statistical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In our experiments, we first study the performance of our tests as the horizon T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, we plot both average Type-I error (to empirically validate the Type-I error control guarantee of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1) as well as power under an alternative distribution, for values of T ranging from 10 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This power plot, however, does not paint the whole picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In each simulation, a combination of two factors influences the power of our method: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the intrinsic difficulty of the environment and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the quality of our resampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As an attempt to disentangle these two components, we additionally measure and plot the power of a standard randomization test on data collected via a baseline uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' adaptive assignment algorithm which selects actions uniformly at random from X at each timestep t independently from Ht−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This baseline measurement captures the intrinsic difficulty of the environment: a less powerful test on data gathered by this baseline i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' treatment assignment indicates a more difficult environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Now, while the power plots do illustrate the second factor regarding the quality of our resampling algorithms, this factor can be further decomposed into two more contributing components to paint a clearer picture: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the effective sample size of the resampling procedure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', ideally having weights close to (m + 1)−1) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the diversity of the resamples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', if all resamples are equal—and equal to the original data—then all weights will be (m+1)−1, but our test will be powerless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We disentangle these two components—and thereby give a more complete explanation of the accompanying power plots—by measuring effective sample size, meff := ��m i=0 wq D( ˜D(i)) �2 �m i=0 wq D( ˜D(i))2 , and plotting meff m , the fractional effective sample size, under the alternative at the same increments as in the power plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Type-I error, power, and fractional effective sample size plots, we take m, the number of resamples drawn by our test, to be 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We also present plots showing how the power of our procedure grows with m (taking values 102, 103, and 104) for fixed T = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that all randomization tests performed in our simulations use smoothed p-values (see Remark 2) and thus provably control Type-I error (whose nominal rate is set at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05 in all our experiments) at exactly the nominal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In addition to testing, we also construct approximate confidence and conformal prediction intervals (at nominal miscoverage rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05), using the procedures described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' as these inversion procedures involve non-smoothed p-values, we expect the coverage to be above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' All results are averaged over 1000 independent trials and plotted with ±2 standard error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we only show plots involving the weighted MC version of our test (and its inversion for interval construction) in this section, because for each resampling procedure we consider, our weighted MC test is never outperformed by (in terms of power and length)—and often in fact dominates—the unweighted MCMC test in all of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For analogous plots illustrating results for the unweighted MCMC test, see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Computation We also plot the computation times for each of our resampling algorithms in the various environments and adaptive data assignment algorithms considered here in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In nearly all environments and assignment algorithms, we are able to run a powerful test on datasets of length T = 100 within a matter of minutes (and often just seconds) in terms of serial computation time, and our test is of course embarrassingly parallelizable if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Generally, the computation times for each resampling 19 Contextual stationary non-reactive environment ϵ-greedy, LinUCB Contextless stationary non-reactive environment ϵ-greedy, UCB Contextless C-stationary strongly non-reactive environment ϵ-greedy, UCB Contextual C-stationary strongly non-reactive environment ϵ-greedy, LinUCB Markov decision process ϵ-greedy Q-learning, greedy Q-learning Table 1: Table of environments and adaptive assignment algorithms considered in each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' procedure scale roughly linearly with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' All code for our simulations is publicly available at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='com/Yashnair123/RTs-for-AdaptiveData.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Environments and adaptive assignment algorithms We briefly describe the environments in and adaptive assignment algorithms for which we conduct our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We consider a total of five different environments: two are examples of a stationary non-reactive environment (Environment 3)—one with con- texts and one without contexts—two are instantiations of a C-stationary strongly non-reactive environment (Environment 5)—again, both contextual and contextless—and the last is an instance of an MDP (Envi- ronment 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' recall that Environments 3 and 5 are special cases of Environment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The adaptive assignment algorithms we consider in these environments are summarized in Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' note that, in each environment, we consider both a randomized and a deterministic adaptive assignment algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now briefly describe each of these adaptive assignment algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Q-learning (Watkins, 1989) maintains an estimate of the state-action value Q function and selects actions at each timestep based on the current estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we consider one version in which this action selection is (deterministically) greedy and one in which it is ϵ-greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The ϵ-greedy algorithm in the contextless stationary non-reactive environment and contextless C-stationary strongly non-reactive environment both, at each timestep, determine the empirically best action and then select an action ϵ-greedily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the contextual stationary non-reactive environment, the ϵ-greedy algorithm behaves similarly by maintaining a linear regressor Lx for each action x ∈ X, and progressively updating Lx with the context-response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', input-output) pair (Ct, Yt) when x is selected at time t (upon seeing Ct);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the algorithm selects, at time t after seeing Ct, the action x with highest predicted response by the Lx’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, UCB (Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2002) is a deterministic bandit algorithm which additively inflates each action’s empirical value by a bound on its error with respect to the true value and selects the highest value action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The contextual analogue in which the response depends linearly on the context is LinUCB (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that essentially all of the adaptive assignment algorithms above are typically used in rein- forcement learning as they all enjoy quite low regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' However, in comparison to much of the literature on asymptotic inference in reinforcement learning, which impose clipping constraints on these assignment algorithms that stipulate that action-selection probabilities cannot be too close to 0 or 1 (Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hadad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2021), our procedure makes no such assumptions and even allows for deterministic adaptive assignment algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Conditional independence testing In this section we apply the conditional independence testing framework and corresponding resampling algorithms from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 to two environments: a stationary strongly non-reactive environment (Environ- ment 3) with contexts and one without contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Our simulations in the former environment demonstrate the power gain our framework has over prior work (Pocock and Simon, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shep- hard, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu, 2022) in a stationary environment by incorporating the additional randomization over permutations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, our simulations in the latter environment focus on the problem of testing if a subset of treatments induce the same response and thus demonstrates our test’s power on an inferential test for which, to the best of our knowledge, no prior exact test exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 20 Figure 1: Type-I error (left) and power (right) of randomization tests at fixed m = 100 and varying T in a contextual stationary strongly non-reactive environment on data gathered via ϵ-greedy and LinUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Conditional independence testing with constant g Our first series of simulations is in a contextual stationary strongly non-reactive environment, and involves testing against the null hypothesis H⊥⊥,g with g(Xt) := ∅ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', Yt ⊥⊥ Xt | Ct for all t ∈ [T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While, as mentioned in Remark 3, this setting has been covered in prior work, our framework offers a more powerful test, as our simulation results illustrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, we note that in this setting, the weighted MC and unweighted MCMC tests are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is because all weights in the MC test are (m + 1)−1 and all acceptance ratios in the MCMC test are 1 since our resampling procedures all involve the imitationX distribution which, under a constant g, is the same as sampling, conditionally on the context and response sequences, from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For this reason, we do not plot the fractional effective sample sizes, as they too are all equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The precise environment in which our simulations in this section are conducted has treatment space X = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Letting Ik denote the k × k identity matrix and ⃗1k the length-k vector of all 1’s, we consider null and alternative distributions involving 2-dimensional contexts Ct sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from N �� 1 −1 � , I2 � and whose conditional response distributions are, respectively, given by: Yt | (Ct, Xt) ∼ N(C⊤ t ⃗12, 1) and Yt | (Ct, Xt) ∼ N(C⊤ t ⃗12 + Xt, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the test statistic used is simply the absolute value of the t-test statistic against β2 = 0 in a Normal linear model with design matrix comprising rows of the form (1, Xt, Ct,1, Ct,2) and response vector (Yt)T t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 1 demonstrates that the added diversity of incorporating uniform permutations—as opposed to sam- pling only from the imitationX distribution—indeed results in an increase in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Notably, when comparing our resampling scheme, uniformπ+imitationX, against the imitationX resampling of prior work (Pocock and Simon, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Simon, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bojinov and Shephard, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Ham and Qiu, 2022) we observe that our approach is more powerful than theirs on data collected via an ϵ-greedy treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Perhaps even more strikingly, our approach when applied to LinUCB, a deterministic adaptive assignment algorithm, has high and increasing (with T) power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is in contrast to the usual imitationX resampling of the prior work, which is powerless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We relegate the power curves for increasing m but finite T to Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 as they are all approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 21 Contextual stationary strongly non-reactive environment conditional independence test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 点 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='10 rpe-l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='08 a6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Q9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='uniformiidbaseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCB uniformn+imitationx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='greedy imitationx (priorwork) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUcB imitationx(priorwork) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='greedyuniformn+imitationxFigure 2: Type-I error rate (leftmost) and power (second from left) of the MC randomization test at fixed m = 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='and varying T as well as power at fixed T = 100 and varying m (third from left) and fractional effective sample size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='plots at fixed m = 100 and varying T (rightmost) in a contextless stationary strongly non-reactive environment on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='data gathered via ϵ-greedy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' UCB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Conditional independence testing with non-constant g We now discuss our simulations in the contextless stationary non-reactive setting of Environment 3, given as an example in the beginning of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, we consider a contextless instantiation of a stationary non-reactive environment with action space X = {0, 1, 2} and are testing against the null hypothesis H⊥⊥,g 0 with g(Xt) := I(Xt = 2) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', that Yt ⊥⊥ Xt | Xt ∈ {0, 1} for all t ∈ [T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In this environment, we specify the null and alternative distributions as Yt | Xt ∼ � N(0, 1) if Xt ∈ {0, 1} N(2, 1) if Xt = 2 and Yt | Xt ∼ � � � � � N(0, 1) if Xt = 0 N(3, 1) if Xt = 1 N(2, 1) if Xt = 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In all our simulations, the test statistic S(D) we use is, similar to as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1, simply the absolute value of the t-test statistic for the test against β2 = 0 in a Normal linear model whose design matrix has (1, I(Xt = 0), I(Xt = 2)) as its tth row and (Yt)T t=1 as the response vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 2 summarizes the results in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' First, the Type-I error rates in Figure 2 are controlled at exactly the nominal level, validating the theoretical guarantee of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' With regards to power, we notice that restricted-uniformπ +imitationX sampling performs better on data gathered via ϵ-greedy while combinedπ,X sampling performs better under UCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The fractional effective sample size plots in the same figure offer an explanation for why this is the case: the fractional effective sample sizes under ϵ-greedy are larger with restricted-uniformπ+imitationX sampling than with combinedπ,X and, conversely, under UCB they are larger with combinedπ,X than restricted-uniformπ + imitationX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note here that this testing problem is much harder on data gathered via ϵ-greedy and UCB than on the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline (thus explaining the power gap between the latter and all resampling procedures applied on data gathered by the former two, especially for large T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is because ϵ-greedy and UCB are low-regret adaptive assignment algorithms and thus, under the alternative, will select action 1 very frequently, and all other actions much less frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence the problem of detecting if the response distributions induced by Xt = 0 and Xt = 1 are the same becomes much more challenging, as action 0 is sampled rarely under these two low-regret algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, it is sampled more frequently and at the same rate as action 1 under the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Despite this challenge, however, our method is still able to attain quite high power under both ϵ-greedy and UCB adaptive assignment algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 22 Contextless stationary strongly non-reactive environment conditional independence test Type-l error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 (Altemative) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 50 75 100 50 75 102 103 104 25 50 75 100 T T m T uniform idbaseline e-greedy uniformn+imitationx UCB restricted-uniformn+imitationx E-greedyrestricted-uniformn+imitationx UCB combinedn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='x UCB uniformn+imitationx greedy combinedn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='xFigure 3: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' UCB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Non-stationarity testing In this section, we empirically evaluate our test of non-stationarity on three different environments: the first two are examples of the C-stationary strongly non-reactive setting of Environment 5—one with contexts and one without—and the third is an MDP (Environment 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Testing non-stationarity in a C-stationary strongly non-reactive environment Our simulations in this section are performed on a contextless C-stationary strongly non-reactive environment with action space X = {−1, 1} and Gaussian rewards: the reward distribution for the first T − 1 steps is given by Yt | Xt ∼ N(Xt, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under the null hypothesis HS 0, the reward distribution at the T th timestep is unchanged and hence YT | XT ∼ N(XT , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We analyze power under an alternative distribution that samples YT | XT ∼ N(4XT , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, as we are testing for non-stationarity, our test statistic S is simply a non-conformity score, and we choose it to be the absolute residual: S(D) = |YT − ˆµD(XT )|, where ˆµD is the fitted ordinary least squares (OLS) model to D with an intercept term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 3 shows the results for our simulations in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Indeed the Type-I error plots once again demonstrate the validity our randomization tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In terms of power, Figure 3 shows that cond-imitationπ sampling performs best under an ϵ-greedy treatment assignment and imitationπ performs best under UCB and both attain power close to that of the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline for nearly all values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 3, however, also shows that, with large enough m, re-imitationπ sampling eventually performs better than cond-imitationπ at T = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Combining this information with the fractional effective sample size plots of Figure 3, we thus see that while cond-imitationπ has greater effective sample size than re-imitationπ, it has less diverse samples, leading to a more powerful procedure when m is small (since, in this regime, the diversity plays a greater role), but a (slightly) less powerful one when m is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Testing non-stationarity in a contextual C-stationary strongly non-reactive environment We now describe our simulations in a contextual C-stationary strongly non-reactive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The specific environment we consider in this section involves a action space X = {−1, 1} and 100-dimensional contexts sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from N(⃗1100, I100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The conditional response distribution during the first T − 1 23 Contextless c-stationary strontly non-reactive environment, non-stationarity test Type-l error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='100 Type- 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='050 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='000 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 50 100 25 50 75 102 103 104 25 50 75 T T m T +- uniform iid baseline greedyimitationn E-greedy cond-imitationn UCBimitationn greedyuniformr greedy re-imitationn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBuniformnFigure 4: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy, LinUCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' timesteps is a sparse linear combination of the context vector and is given by Yt | (Ct, Xt) ∼ N(−5Xt + 10 � j=1 Ct,j, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The null conditional response distribution at time T is of course the same as the above whereas the alternative distribution we evaluate power under swaps the effects of the two treatments and is given by YT | (CT , XT ) ∼ N(5XT + 10 � j=1 CT,j, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We regularize the regressor Lx used in the ϵ-greedy adaptive assignment by using Lasso regression with with penalty parameter 10 as opposed to OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the test statistic used is the same non-conformity score as in the previous section, except that we again use Lasso—this time with penalty parameter chosen through 5-fold cross-validation—instead of OLS, as ˆµD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 4 shows the Type-I error, power, and effective sample size curves for our simulations in this envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The Type-I error is again controlled at the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In terms of power, similar conclusions to those drawn in the contextless C-stationary strongly non-reactive environment of the previous section apply here, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, while cond-imitationπ sampling performs best under an ϵ-greedy adaptive assignment at m = 100 (and again exhibits, for large T, power quite close to—and, in fact greater than, for small T—that of the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline) it has essentially the same power as re-imitationπ at both m = 103 and m = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Using the fractional effective sample size plots, we may therefore infer, once again, that while cond-imitationπ sampling has a higher effective sample size than re-imitationπ does under ϵ- greedy, its samples exhibit less diversity than those of re-imitationπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under LinUCB, the contextual analog of the deterministic UCB assignment, however, we see that while imitationπ has quite high power for small to moderate values of T, the power drops for larger T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The fractional effective sample size curve explains that this is caused by a corresponding drop in the effective sample size as T grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We hypothesize that this is due to the extremely uneven action selection exhibited by LinUCB due to its very low regret (even as compared to ϵ-greedy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we discuss why this low regret and the uneven action selection it causes—which we again emphasize is extreme in the case of LinUCB, even in comparison to ϵ-greedy—renders the testing problem harder in the paragraph below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, we leave the problem of developing a powerful resampling scheme for deterministic algorithms like LinUCB in this type of contextual C-stationary strongly non-reactive environment to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 24 Contextual c-stationary strontly non-reactive environment, non-stationarity test Type-l error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='100 Type- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='075 ae abe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='050 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='000 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 50 100 25 @5 75 102 103 104 25 50 75 m T +-uniformidbaseline greedy imitationr greedy cond-imitationm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCBimitationn greedy uniformr greedy re-imitationn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCB uniformnLastly, we hypothesize that the shift in relative power of the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline (from the least powerful adaptive assignment-resampling algorithm combination for small T to the most powerful for large T) is again an artifact of the low-regret properties of ϵ-greedy and LinUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This low regret results in the outlier context- action-response triple sampled at the T th timestep to often have action agreeing with the action taken during most of the first T − 1 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, for small T, these first T − 1 context-treatment-response triples may outweigh the effect of the outlier at time T when training ˆµD via Lasso with cross-validation, thereby resulting in better outlier detection than that under the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' base, wherein only around half of the first T − 1 timesteps (and thus very few, in total) will have action agreeing with that at the T th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The effect however, is diminished with large T, most likely because in that regime even half of the data generated via Yt’s true (null) conditional distribution, which agrees with the action taken at time T, is enough to offset the effect of the outlier in training ˆµD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, the remaining timesteps whose corresponding action differs from the one taken at time T—of which there are more under the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline than under ϵ-greedy (and many more under the baseline than under LinUCB) for large T—may allow for a better fit ˆµD through more effective learning of the dependence of Yt on Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As noted above, this may also explain the difficulty in attaining high power for large T under LinUCB in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We empirically validate this hypothesis in Figure 24 of Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 by showing that the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline exhibits precisley this same relative performance, with respect to T, in comparison to a biased i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' assignment algorithm that selects action −1 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='9 and otherwise selects action 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Testing non-stationarity in an MDP Our final set of hypothesis testing simulations is in an MDP, where recall that Yt = Ct+1 and hence we drop the notation Yt and refer to Ct as states instead of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='9 The precise specifications of environment involve a state space of C = {0, 1, 2}, action space X = {−1, 1}, and transition kernel during the first T − 1 transitions given by Ct+1 | (Ct, Xt) ∼ � Ct + Xt (mod 3) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='95 Ct − Xt (mod 3) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under the null hypothesis, the transition distribution at time T remains the same as above, but under the alternative, we swap the role of the two actions so that the alternative distribution is given by CT +1 | (CT , XT ) ∼ � CT − XT (mod 3) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='95 CT + XT (mod 3) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the test statistic that we use in these simulations trains a decision tree classifier on the data D and outputs the negative log likelihood loss of the trained model on the triple (CT , XT , CT +1) at time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 5 summarizes the results for our simulations in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Once again, Type-I error is controlled at exactly the nominal level, as guaranteed by our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In a departure from the results of the previous two sections (most likely due to the sequential dependence in this environment not present in the last two as well as the modified sampling process that it requires), uniformπ resampling has the greatest power for Q-learning with ϵ-greedy action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, both uniformπ and imitationπ resampling perform similarly well for Q-learning with greedy action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Again, using both the fractional effective sample size plots and power plots with fixed T but varying m, we see that under ϵ-greedy, re-imitationπ resampling has lower effective sample size than uniformπ, but higher diversity, leading it to perform worse for smaller m but better for large m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we note that, in comparison to the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline10, uniformπ 9As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, the dataset we consider in these simulations is the usual dataset D along with one final action taken at time T + 1: ((C1, X1, C2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , CT +1), XT +1) as this allows us to simply permute the state-action pairs (Ct, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 10In contrast to the last section, the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' data used in this section is not actually gathered in the same environment as the other adaptive assignment algorithms, and in particular, the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' data is not MDP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Rather, the data comprises state-action-next state triples (C, X, C′) sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from a distribution in which both C and X are independent and uniform over C and X respectively, and C′ is sampled from the transition distribution described above, conditional on (C, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 25 Figure 5: Type-I error rate (leftmost) and power (second from right) of the MC randomization test at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy and greedy Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and imitationπ are very competitive under the greedy Q-learning adaptive assignment, and imitationπ also exhibits quite high power under ϵ-greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Constructing confidence and prediction intervals In this section we apply our framework to constructing confidence and prediction intervals using the inversion procedures described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, we plot both the coverage of the intervals as well as their average length, averaged over 1000 trials for varying T at a fixed number of MC samples m = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For construction of conformal intervals, we also apply the sample sharing described in Remark 7 (in addition to the standard gridding procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3) at both m = 10 and m = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we note that some of the resampling procedures in certain environments and adaptive assignment algorithms considered in this section exhibit overcoverage, in contrast to the last section, in which all tests attained Type-I error control at exactly the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We remind the reader that this is due to the fact that the gridding procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 is based on the (approximate) inversion of a conservative test, rather than that of the smoothed test described in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Confidence intervals We now discuss our simulations constructing confidence intervals using the gridding procedure described at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We consider essentially the same environment as the contextless stationary strongly non-reactive environment discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, except that here we have Yt | Xt ∼ � � � � � N(0, 1) if Xt = 0 N(b0, 1) if Xt = 1 N(2, 1) if Xt = 2 , with b0 = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we construct a confidence interval for the location difference between Yt | (Xt = 0), and Yt | (Xt = 1), which simply corresponds to b0, using a gridding set Y′ = [−1 : 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 6 summarizes the results for these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We note there is no evidence in our simulations that the necessary (and in fact rather coarse) gridding of Y results in any undercoverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In terms of length, our results align with the power results presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, under an ϵ-greedy adaptive assignment, 26 MDP, non-stationarity test Type-l error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='100 Type-l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='075 Average Average (Alter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='050 mim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 50 75 100 25 50 75 100 102 103 104 25 50 75 100 T T m T +-uniformiidbaseline greedy cond-imitationr + -greedy uniformn greedy uniformr greedyimitationn greedy imitationn greedy re-imitationnFigure 6: Coverage and average length of confidence intervals for location difference between Yt | (Xt = 0), and Yt | (Xt = 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', b0 = 4) using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' restricted-uniformπ+imitationX sampling produces the shortest average length intervals, whereas under UCB, combinedπ,X resampling does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Conformal prediction intervals Finally, we discuss our simulations constructing conformal prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The environment considered in this section is the same as contextless C-stationary strongly non-reactive environment considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 except that we do not have access to YT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' it is the quantity for which we construct the conformal prediction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Our simulations in this section use the gridding procedure described at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 and assess the benefit of the sharing of samples described in Remark 7 and Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Without sample sharing, we fix m = 100 and set Y′ = [−5 : 5] and plot coverage and length at varying T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' With sample sharing, we use the same gridding set Y′ and study both m = 10 and m = 100 number of MC samples11 and varying T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As discussed in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 this sample sharing results in non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' draws because the resamples generated corresponding to y1 ∈ Y′ may come from (and indeed do in our simulations) a different distribution than those sampled according to y2 ∈ Y′ for y1 ̸= y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, following Remark 8 (and as discussed in further detail in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1), we choose a subset of permutations Σ to be the set of m + 1 permutations swapping 0 and i for each i ∈ [0 : m] and employ the test of Algorithm 1 with weights w˜q,Σ D ( ˜D(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figures 7 and 8 summarize our simulation results constructing approximate conformal prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 7 illustrates both coverage and length as T grows by simply using the standard gridding procedure described at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' With regards to average length, our results mirror those in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1, in which cond-imitationπ resampling is best for an ϵ-greedy adaptive assignment, and imitationπ is best for UCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In Figure 8, we observe that any undercoverage, attributable to the gridding of Y, is relatively minor, and should be fixable by using a finer-resolution grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the reason that the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline tends to exhibit the greatest undercoverage is most likely explained by that fact that we are not smoothing in this set of experiments, so that whereas all other procedures are conservative, the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, we see essentially the same ranking of resampling procedures, although re-imitationπ does perform slightly better than cond-imitationπ when 100 MC samples are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, the shortest 11For each y ∈ Y′, m samples are generated, but all m|Y′| samples are used to determine the membership of y ∈ Y′ in the prediction interval 27 Confidence interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content="98 8 96'0 Coverage Average Length 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='86 20 40 60 100 80 20 40 60 80 100 T T uniformiidcomparator greedy uniformn+imitationx UCB restricted-uniform,+imitationx E-greedy restricted-uniformn+imitationx UCB combinedn,x UCB uniform,+imitationx greedy combinedr,xFigure 7: Coverage and average length of conformal prediction intervals for YT using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 8: Coverage and average length of approximate conformal prediction intervals, with sample sharing, for YT using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 28 Contextless C-stationary strongly non-reactive environment, conformal prediction interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='98 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='96 Average Length Cov 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content="92 ferage 06'0 A 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='86 20 40 09 80 100 uniformiidbaseline E-greedyimitationn E-greedy cond-imitationn UCB imitationn E-greedy uniformn E-greedy re-imitationm UCB uniformnContextual c-stationary strongly non-reactive environment, conformal prediction interval, shared samples 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 QT 10 (00T =) Coverage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='95 Coverage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='95 T 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='90 Average N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='85 2550 25 50 75 100 [0] 25 50 75 100 [0] 75 100 25 5075 100 T T T T uniform iid baseline E-greedy cond-imitationn E-greedy uniformn greedy imitationn UCB imitationn UCB uniformr E-greedy re-imitationnaverage length curves in Figure 8 using m = 10 are quite competitive in comparison to their counterparts in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, imitationπ under UCB produces only slightly wider intervals with m = 10 and sample sharing than with m = 100 and no sample sharing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' for ϵ-greedy, cond-imitationπ performs essentially the same in both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This demonstrates that, by utilizing sample sharing, we are able to obtain conformal prediction intervals of nearly the same length at a fraction of the computational cost (because in the left hand column of Figure 8 we generate a total of 110 samples, whereas in Figure 7, we generate 100 samples for each y ∈ Y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Indeed, as illustrated by Figures 21 and 22 of Appendix G, the computation time required by the procedure using m = 10 and shared sampling is, for most resampling schemes and adaptive assignments, more than 5 times faster than its m = 100 non-sample sharing counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 6 Discussion In this paper, we study the problem of performing various challenging inferential tasks on adaptively collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Through the development of a weighted MC randomization test (along with the novel application of the unweighted MCMC randomization test of Besag and Clifford (1989)), we show that such tasks can be performed with exact Type-I error control with a great degree of flexibility in the choice of resampling procedure as long as the proportionality hypothesis H∝ 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The question, however, of how best to perform these tests still remains, and is an interesting one for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, while we have discussed some preliminary empirical results demonstrating powerful resampling algorithms in a number of different challenging environments, there are a number of interesting and potentially fruitful directions forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, as mentioned in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, one important direction for future work is to develop resampling procedures that are powerful for deterministic algorithms like LinUCB in a contextual C-stationary strongly non-reactive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' But more generally, it may be interesting to formally investigate any commonalities between powerful resampling algorithms in different environments in order to develop a more principled method for deciding which procedure to apply to a particular problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', combination of adaptive assignment algorithm, environment, and inferential task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Along this same vein of how best to resample, it may be useful to also consider different resampling procedures which incorporate non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Utilization of these non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling schemes under the unweighted MCMC randomization testing framework has the potential to be especially advantageous as the full MCMC test will still require only O(m) number of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, as discussed in Remark 8 and and Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1, when applying such sampling schemes to the weighted MC test, we may generalize and choose a subset Σ of the full permutation set Π[0:m] different than Σswap, and condition only on the data permutations induced by these sets rather than the full set of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, one final direction for future work that may also be of interest is to investigate how the power of the weighted MC test changes under different choices of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, while we expect the test to be more powerful for choices of Σ which are larger, we also expect such larger choices to require a greater amount of comptutation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, it may be worthwhile to study this tradeoff between the size of the sets Σ and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Just as in the potential investigation of more powerful resampling procedures mentioned above, exploring if there are more well-structured connections between not-too-large choices of Σ that yield a powerful test and the specific problem at hand may allow for the development of a more coherent theory—or at least a more formal set of guidelines—for how best to select/develop a resampling algorithm for any given problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' References Ronald Aylmer Fisher et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Testing goodness-of-fit and conditional independence with approxi- mate co-sufficient sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Annals of Statistics, 2022+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To Appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Clara Fannjiang, Stephen Bates, Anastasios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Angelopoulos, Jennifer Listgarten, and Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Conformal prediction for the design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' CoRR, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='03613, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='org/ abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='03613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Dae Woong Ham and Jiaze Qiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hypothesis testing in sequentially sampled data: Adaprt to maximize power beyond iid sampling, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='02430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Tianhao Li, Zhishun Wang, Wei Lu, Qian Zhang, and Dengfeng Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Electronic health records based rein- forcement learning for treatment optimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Information Systems, 104:101878, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Kelly Zhang, Lucas Janson, and Susan Murphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Statistical inference after adaptive sampling in non- markovian environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='07098, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Yao Zhang and Qingyuan Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' What is a randomization test?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 10980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 33 A Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 We consider the case of discrete data D and discrete resamples ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) and apply Bayes’ theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Define d = (d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dm) to be be some list of data sets, d−0 to be the list (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dm), D−0 to denote the list ( ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)), and orb(d) := {dπ : π ∈ Π[0:m]}, where dπ := (dπ(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we also use (d−0)π to denote the permuted list (dπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m)) for any π ∈ Π[0:m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then, by Bayes’ theorem, we have that P (D = di|D ∈ orb(d)) = P (D ∈ orb(d)|D = di) f(di) � distinct dj∈d P (D ∈ orb(d)|D = dj) f(dj) = P � D−0 ∈ {(d−0)π : π ∈ Π[0:m] such that dπ(0) = di}|D = di � f(di) � distinct dj∈d P � D−0 ∈ {(d−0)π : π ∈ Π[0:m] such that dπ(0) = dj}|D = dj � f(dj) = |{(d−0)π : π ∈ Π[0:m] such that dπ(0) = di}| �� k∈[0:m]\\{i} q(dk|di) � f(di) � distinct dj∈d |{(d−0)π : π ∈ Π[0:m] such that dπ(0) = dj}| �� k∈[0:m]\\{j} q(dk|dj) � f(dj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' where we say that d ∈ d if d is an element of the list d and where the last equality follows from the conditional i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='-ness of the resamples given D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Letting mdi(d) denote the number of times di appears in the list d we have that |{(d−0)π : π ∈ Π[0:m] with dπ(0) = di}| = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (mdi(d) − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � distinct d∈d not equal to di md(d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' = mdi(d) · m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � distinct d∈d md(d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and hence, P (D = di|D ∈ orb(d)) = mdi(d)f(di) � k∈[0:m]\\{i} q(dk|di) �m j=0 f(dj) � k∈[0:m]\\{j} q(dk|dj) = mdi(d) ˆf(di) � k∈[0:m]\\{i} q(dk|di) �m j=0 ˆf(dj) � k∈[0:m]\\{j} q(dk|dj) = mdi(d)wq d(di), where the second-to-last inequality is because of the proportionality hypothesis H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Equivalently, we can think of D’s conditional distribution given orb(D) as a draw from the m + 1 elements of D, with weight on each element given by the wq D function applied to that element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Defining the set S := {S( ˜D) : ˜D ∈ D}12, S(D) can be viewed as a draw from the weighted distribution on S with the total weight of each element S( ˜D(i)) equal to � j:S( ˜ D(j))=S( ˜ D(i)) ˆf( ˜D(j)) � k∈[0:m]\\{j} q( ˜D(k)| ˜D(j)) �m j′=0 ˆf( ˜D(j′)) � k′∈[0:m]\\{j′} q( ˜D(k′)| ˜D(j′)) = � j:S( ˜ D(j))=S( ˜ D(i)) wq D( ˜D(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, note that P(p ≤ α) = P � m � i=0 wq D( ˜D(i))1[S( ˜D(i)) ≥ S(D)] ≤ α � = P � �S(D) > inf{s ∈ S : � ˜ D∈D:S( ˜ D)≤s wq D( ˜D) ≥ 1 − α} � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 12Just as above, we say that ˜D ∈ D if ˜D is an element of the list D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 34 Then, we have that P � �S(D) > inf{s ∈ S : � ˜ D∈D:S( ˜ D)≤s wq D( ˜D) ≥ 1 − α} ������ orb(D) � � ≤ α, (11) since 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the infimum in equation (11) is the (1 − α)th conditional quantile of the weighted distribution over S and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' as noted above, S(D) is conditionally a draw from this weighted distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Inequality (11) then also holds unconditionally, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' B Conditional independence testing in a stationary MDP In this section, we give a proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 regarding the restricted permutation set in Environment 4: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Let πi be any permutation in Π4 := {π ∈ Π[T ] : Yπ(t) = Cπ(t+1), ∀t ∈ [T − 1], and Cπ(1) = C1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then ˆf( ˜D(i)) = T � t=1 PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) = �T t=1 P(Yπi(t)|Cπi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xπi(t))PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) �T t=1 P(Yt|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt) because of equation (6) = P(Cπi(1)) �T t=1 P(Cπi(t+1)|Cπi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xπi(t))PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) P(C1) �T t=1 P(Ct+1|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt) since πi ∈ Π4 = P(Cπi(1)) �T t=1 P(Cπi(t+1)|Cπi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) t )PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) P(C1) �T t=1 P(Ct+1|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt) by H⊥⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g 0 and that g( ˜X(i) t ) = g(Xπi(t)) = P( ˜C(i) 1 ) �T t=1 P( ˜C(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) t )PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) P(C1) �T t=1 P(Ct+1|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt) by equation (8) = 1 P(C1) �T t=1 P(Ct+1|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Xt) f( ˜D(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' due to equation (1) and so the proportionality hypothesis is satisfied with K = 1 P(C1) �T t=1 P(Ct+1|Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' C Inference in a general adaptive data collection process In this section, we relax all structural assumptions made in Section 3 and assume that the data are gathered according to any adaptive data collection process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', we make no assumptions on the joint distribution of (Ct, Xt, Yt)T t=1 other than that the sequence (Xt)T t=1 is non-anticipating with respect to the filtration Ft := σ (Ht−1 ∪ {Ct})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The hypothesis we are interested in testing is related to that discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1, except here it involves sequences of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In fact, we consider T functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , gT , where gt takes as input the sequence of first t actions X1:t := (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Xt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the null hypothesis we are interested in testing is that of simultaneous independence, conditional on these g-evaluations: H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 : [Ct ⊥⊥ X1:t−1 | (C1:t−1, gt−1(X1:t−1), Y1:t−1)] and [Yt ⊥⊥ X1:t | (C1:t, gt(X1:t), Y1:t−1)] , ∀t ∈ [T], where we define C1:0 = X1:0 = g0(X1:0) = Y1:0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 35 Informally, H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 states that the actions Xt only influence future contexts and responses via their values filtered through the functions gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As noted in Remark 4, the setting in which all gt are constant has been covered in prior work by Bojinov and Shephard (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' however, to the best of our knowledge, the case of non-constant gt is novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now describe and prove the validity of the testing procedure for the null hypothesis H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Suppose that the resampling distribution q randomizes only the action sequence X1:T conditional on (gt(X1:t))T t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, in each resample ˜D(i), we have that � ˜C(i) t , gt( ˜X(i) 1:t), ˜Y (i) t � = (Ct, gt(X1:t), Yt) , ∀t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (12) Then the null hypothesis H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 implies the proportionality hypothesis H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We have that ˆf( ˜D(i)) = T � t=1 PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1) = �T t=1 P( ˜Y (i) t | ˜C(i) 1:t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt( ˜X(i) 1:t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Y (i) 1:t−1)PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1)P( ˜C(i) t | ˜C(i) 1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt−1( ˜X(i) 1:t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Y (i) 1:t−1) �T t=1 P(Yt|C1:t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt(X1:t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Y1:t−1)P(Ct|C1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt−1(X1:t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Y1:t−1) equation (12) = �T t=1 P( ˜Y (i) t | ˜C(i) 1:t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt( ˜X(i) 1:t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Y (i) 1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) 1:t)PA( ˜X(i) t | ˜C(i) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜H(i) t−1)P( ˜C(i) t | ˜C(i) 1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt−1( ˜X(i) 1:t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Y (i) 1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜X(i) 1:t−1) �T t=1 P(Yt|C1:t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt(X1:t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Y1:t−1)P(Ct|C1:t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' gt−1(X1:t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Y1:t−1) due to H⊥⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 = �T t=1 P( ˜Y (i) t | ˜C(i) 1:t, ˜X(i) 1:t, ˜Y (i) 1:t−1)PA( ˜X(i) t | ˜C(i) t , ˜H(i) t−1)P( ˜C(i) t | ˜C(i) 1:t−1, ˜X(i) 1:t−1, Y (i) 1:t−1) �T t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) = 1 �T t=1 P(Yt|C1:t, gt(X1:t), Y1:t−1)P(Ct|C1:t−1, gt−1(X1:t−1), Y1:t−1) f( ˜D(i)), which satisfies the proportionality hypothesis with K = 1 �T t=1 P(Yt|C1:t,gt(X1:t),Y1:t−1)P(Ct|C1:t−1,gt−1(X1:t−1),Y1:t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' One concrete setting in which the above randomization procedure could be used to test against the hypothesis H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 is in a completely general mobile health setting in which no environmental assumptions are made and there are only two treatments: a control and an experimental treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' A hypothesis which may be of interest to test is that, conditional on the prior sequence of contexts and responses, each of the next response and context is independent of the action sequence given the number of experimental treatments taken up until that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In such a setting, one could use our framework under the above randomization scheme to test the null hypothesis H⊥⊥,g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=',gT 0 where gt(X1:t) is equal to the number of times the experimental treatment appears in the treatment sequence X1:t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', gt(X1:t) = �t s=1 Xs where X = 1 denotes the experimental treatment and X = 0 denotes the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' D Inference in scale families In this section, we describe how the test discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 can also be used for inference in semi- parametric scale families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Instead of assuming that each reward distribution is a member of the same location family, it may be more natural in some situations to assume a semiparametric scale family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, letting θx now denote action x’s scale parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', we assume that Yt | Xt = x ∼ h0(y/θx)13), we may instead test against the null 13Recall that we only consider discrete random variables, per Remark 1, and hence no Jacobian is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 36 hypothesis HScale,δ,x,x′ that θx′ θx = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To do so, we simply revise the reward-modification trick discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 for the location family to instead replace Yt with Yt · δ · 1[Xt = x] and again perform the usual test for conditional independence from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 using g(Xt) = � {x, x′} if Xt ∈ {x, x′} Xt otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Similar to as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1, the above test can be inverted to construct (simultaneous) confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' E Non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling and sharing samples In this section, we discuss how to handle non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling procedures and how this more general procedure can be used to share samples between different values y ∈ Y in the construction of conformal prediction regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling In this section, we describe a more general version of the weighted MC randomization test of Algorithm 1 that can be used in the setting of non-conditionally-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling, as discussed in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In particular, the Remark states that, letting ˜q denote the joint conditional distribution of the resamples ( ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)), one may redefine the weights to be w˜q,Σ D ( ˜D(i)) = ˆf( ˜D(i)) � π∈Σ:π(0)=i ˜q((D−0)π| ˜D(i)) � j∈[0:m] ˆf( ˜D(j)) � π′∈Σ:π′(0)=j ˜q((D−0)π′| ˜D(j)) , (13) and the p-value p := m � i=0 w˜q,Σ D ( ˜D(i))1[S( ˜D(i)) ≥ S(D)] will be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Recall that the key idea behind Algorithm 1—and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1—was to condition on the event ( ˜D(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) ∈ {(dπ(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m)) : π ∈ Π[0:m]} for some list d = (d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dm) and to apply Bayes’ theorem as well as the proportionality hypothesis to derive the weight formula (equation (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The entire permutation set Π[0:m] need not, however, be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Our generalization involves using any arbitrary subset of permutations Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' when Σ is small, our method is computationally tractable for non-conditionally- i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More specifically, define pseudo-orbΣ(d) := {(dπ(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m)) : π ∈ Σ} to be the Σ-pseudo-orbit14 of the list d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then, by conditioning on pseudo-orbΣ(D), we will be able to show that the weighting given by equation (13) will be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We first prove a result similar to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 in the case of distinct data D and resamples ˜D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) using exactly the same proof strategy: Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Let Σ be any subset of the full set of permutations on [0 : m], Π[0:m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Define Σi to be {π ∈ Σ : π(0) = i} and assume that ˜q is such that ˜D(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m) are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then, with weights w˜q,Σ D,i( ˜D(i)) := ˆf( ˜D(i)) � π∈Σi ˜q((D−0)π| ˜D(i)) �m j=0 ˆf( ˜D(j)) � π′∈Σj ˜q((D−0)π′| ˜D(j)) , we have that p := m � i=0 w˜q,Σ D,i( ˜D(i))1[S( ˜D(i)) ≥ S(D)] stochasically dominates the uniform distribution under H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 14We use the prefix “pseudo” since Σ need not be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 37 Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We give a sketch of the proof as much of it is the same as that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Using Bayes’ theorem, we have that P(D = di|( ˜D(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) ∈ pseudo-orbΣ(d)) = P(( ˜D(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) ∈ pseudo-orbΣ(d)|D = di)f(di) �m j=0 P(( ˜D(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m)) ∈ pseudo-orbΣ(d)|D = dj)f(dj) = f(di) � π∈Σi P((D−0) = (dπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m))|D = di) �m j=0 f(dj) � π′∈Σj P((D−0) = (dπ′(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ′(m))|D = dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, conditional on pseudo-orbΣ(D), S(D)’s distribution is indeed the weighted distribution on the multiset {S( ˜D) : ˜D ∈ D} with weight of the ith element equal to w˜q,Σ D,i( ˜D(i)) under H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The rest of the proof, in this setting of distinct resamples, follows in precisely the same manner as that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We now extend to the case of repeated samples via a coupling argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Define D′ := D × [0 : m] so that, for each point d in D it has m + 1 “clones” in D′ given by (d, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (d, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We also define a density f ′ on D′ given by f ′((d, i)) := (m + 1)−1f(d) for all d ∈ D, i ∈ [0 : m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Lastly, we extend the test statistic S to a test statistic S′ on D′ which also takes each clone to the value of its original: S′((d, i)) = S(d) for all d ∈ D, i ∈ [0 : m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We will couple p to a random variable p′ obtained by a procedure on the space D′ which draws distinct elements ˜D′(i), to be defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We call the original process (that may draw repeated elements and acts only on D) P and the coupled process P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The observed dataset that the coupled process P′ will see and use is a cloned version (D, j) of the true observed data D, where j is sampled uniformly at random from the set [0 : m];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' importantly P′ treats this cloned dataset D′ := (D, j) as though it were the observed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we use ˜q′ to denote the conditional resampling distribution of P′, induced by ˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' That is, for a list D′ −0 := ( ˜D′(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D′(m)) of resamples in D′, ˜q′(D′ −0|D′) denotes the joint conditional probability of P′ having obtained the list of resamples D′ −0 given that it observed the dataset D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Algorithm 2 describes how P′ is defined with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Algorithm 2: Coupling of P′ to P Input: P′’s observed dataset D′ := (D, j) where j ∼ Unif([0 : m]) 1 ˜D′(0) ← D′ 2 D′(0) ← ( ˜D′(0)) 3 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m do 4 ˜D(i) ← ith element of D in the process P 5 j ← Unif({j′ ∈ [0 : m] : ( ˜D(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' j′) ̸∈ D′(i−1)}) 6 ˜D′(i) ← ( ˜D(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' j) 7 D′(i) ← concatenation of D′(i−1) with ˜D′(i) 8 D′ ← D′(m) Output: p′ := �m i=0 1[S′( ˜D′(i)) ≥ S′(D′)]f ′( ˜D′(i)) � π∈Σi ˜q′((D′ −0)π| ˜D′(i)) �m j=0 f ′( ˜D′(j)) � π′∈Σj ˜q′((D′ −0)π′| ˜D′(i)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' At a high level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' the coupled process P′ draws distinct elements by sampling an element’s clone number j uniformly at random from all remaining yet-unselected values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' using these distinct resamples, it then calculates a p-value in precisely the same way as Algorithm 1 with weights w˜q,Σ D′ ( ˜D′(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Since the draws of P′ are guaranteed to be distinct, Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 will guarantee p′’s stochastic domination of the uniform distribution as long as f ′ is the true density of D′, the observed data for P′ (since ˜q′ is the conditional resampling distribution and because f ′ is trivially proportional to itself, thereby satisfying the proportionality hypothesis in this coupled model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' For d′ ∈ D′, let C(d′) denote the second component of d′ 38 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', the clone number of d′) and d be the first component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', the uncloned version of d′ in D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' then we see that f ′ is D′’s true density: P(D′ = d′) = P(C(D′) = C(d′)|D = d)P(D = d) = P(C(D′) = C(d′))P(D = d) since the clone number j is sampled independently of D = (m + 1)−1f(d) because j ∼ Unif([0 : m]) to clone D = f ′(d′) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We complete the coupling by showing that ˜q′((d′ −0)π|d′ i) = κ˜q((d−0)π|di), ∀i ∈ [0 : m], π ∈ Σi for some κ > 0 depending on neither i nor π, (14) where d is some list (d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dm) of elements in D and d′ := (d′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , d′ m) is a list of elements of D′ comprising distinct clones of the di (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', the first component of d′ i is equal to di, and each element of d′ is distinct), (d′ −0)π := (d′ π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , d′ π(m)), and (d−0)π := (dπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , dπ(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To see why this is true, note that if P′ observes d′ i as the original dataset, then P must have seen its uncloned version: di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, applying the cloning number function C defined above to (d′ −0)π elementwise, we have that ˜q′((d′ −0)π|d′ i) = P(D′ −0 = (d′ −0)π|D′ = d′ i) = P(C(D′ −0) = C((d′ −0)π), D−0 = (d−0)π|C(D′) = C(d′ i), D = di) = P(C(D′ −0) = C((d′ −0)π)|D−0 = (d−0)π, D′ = d′ i)P(D−0 = (d−0)π|D = di) as D−0 ⊥⊥ C(D′) | D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To see that this is proportional to ˜q((d−0)π|di) (which is precisely the second term in the last line above),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' let md((d−0)π) denote the number of elements in (d−0)π with first component equal to d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and observe that we may write the first term in the last line above as P(C(D′ −0) = C((d′ −0)π)|D−0 = (d−0)π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' D′ = d′ i) = � � � distinct d∈d not equal to di 1 �md((d′ −0)π) k=1 (m + 1 − k + 1) � � · 1 �mdi((d′ −0)π) k=1 (m + 1 − k) = � � � distinct d∈d not equal to di (m + 1 − md((d′ −0)π))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � � · (m − mdi((d′ −0)π))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Now note that md((d′ −0)π) does not depend on the choice of π ∈ Σi and so we need only consider the permutation π∗ i ∈ Σi given by π∗ i : k �→ � � � � � i if k = 0 k − 1 if 0 < k ≤ i k if i < k ≤ m so that, for any π ∈ Σi, md((d′ −0)π) = md((d′ −0)π∗ i ) = md(d′ −i), where d′ −i := (d′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , d′ i−1, d′ i+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , d′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, combining this fact with what has been shown above, gives that P(C(D′ −0) = C((d′ −0)π∗ i )|D−0 = (d−0)π∗ i , D′ = d′ i) = � � � distinct d∈d not equal to di (m + 1 − md(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � �·(m − mdi(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Note that the right-hand side above depends on di only through its value, not its subscript, and hence the equation above takes the same value for any j ̸= i for which dj = di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the case of distinct i, j with di ̸= dj, 39 the above gives that P(C(D′ −0) = C((d′ −0)π∗ i )|D−0 = (d−0)π∗ i , D′ = d′ i) = � � � distinct d∈d equal to neither di nor dj (m + 1 − md(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � � · (m + 1 − mdj(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m − mdi(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' = � � � distinct d∈d equal to neither di nor dj (m + 1 − md(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � � · (m + 1 − mdj(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m − mdi(d′ −i))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' = � � � distinct d∈d equal to neither di nor dj (m + 1 − md(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � � · (m − mdj(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1 − mdi(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' = � � � distinct d∈d not equal to dj (m + 1 − md(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' � � · (m − mdj(d′ −j))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' = P(C(D′ −0) = C(d′ −j)|D−0 = d−j, D′ = d′ j) = P(C(D′ −0) = C((d′ −0)π∗ j )|D−0 = (d−0)π∗ j , D′ = d′ j), as desired, where the third equality follows from the fact that mdj(d′ −j)+1 = mdj(d′ −i) for any pair i, j with di ̸= dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' By the same argument made for π∗ i , the last line of the above remains true if we replace π∗ j with any π ∈ Σj and thus, we have shown that not only does P(C(D′ −0) = C((d′ −0)π)|D−0 = (d−0)π, D′ = d′ i) not depend on the choice of π ∈ Σi, for any given i, but it also does not depend on i ∈ [0 : m], and hence we have the desired proportionality of equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, since f ′( ˜D′(i)) = (m+1)−1f( ˜D(i)) by definition, and ˆf( ˜D(i)) = Kf( ˜D(i)) ∀i ∈ [0 : m] for some K not depending on i under the proportionality hypothesis, we have that ˆf( ˜D(i)) = K(m + 1)f ′( ˜D′(i)), ∀i ∈ [0 : m] and hence the two are proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, we have that �m i=0 1[S′( ˜D′(i)) ≥ S′(D′)]f ′( ˜D′(i)) � π∈Σi ˜q′((D′ −0)π| ˜D′(i)) �m j=0 f ′( ˜D′(j)) � π′∈Σj ˜q′((D′ −0)π′| ˜D′(j)) = �m i=0 1[S( ˜D(i)) ≥ S(D)] ˆf( ˜D(i)) � π∈Σi ˜q((D−0)π| ˜D(i)) �m j=0 ˆf( ˜D(j)) � π′∈Σj ˜q((D−0)π′| ˜D(j)) and so p′ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This combined with the fact shown above that p′ is a valid p-value implies the desired validity of p under H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Sharing samples between y ∈ Y We now show how the above framework can be used to share samples between different grid values y ∈ Y as discussed in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We describe this sample sharing in the absence of the rounding procedure of Remark 7 which considers only a small grid Y′ ⊆ Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', we will obtain samples for each y in the full support Y and share these samples amongst all other elements of Y), but both of these methods can be combined to construct conformal prediction regions even more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In more detail, recall that the naive interval construction described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 runs |Y| independent non-stationarity tests, using our weighted MC randomization testing framework for conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' re- samples using q, at each y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To formalize this notion, set (D[1:T −1], CT , XT ) to denote the entire dataset except the last response and let Uy denote the exogenous randomness used by the weighted MC randomization test when determining if y is in the acceptance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then we can define an acceptance function ϕ(Uy, (D[1:T −1], CT , XT ), y) which is 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', accepts) if the test, using Uy as exogenous random- ness, states that y is in the acceptance region upon seeing (D[1:T −1], CT , XT ), and is otherwise 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', rejects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the naive gridding procedure, the Uy are all jointly independent and are distributed identically 40 to the exogeneous randomness U that is used in the usual non-stationarity test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' As such, it is clear that E[ϕ(UYT , (D[1:T −1], CT , XT ), YT )] ≥ 1 − α due to the validity of the non-stationarity test proved in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 and so the corresponding prediction region attains coverage at least that of the nominal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The Uy, however, need not be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Instead, for example, if we have Uy = U for all y ∈ Y, where, again, U is the independent exogenous randomness used by the non-stationarity test, then our prediction region would once again control miscoverage at the nominal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' More generally, as long as each Uy (and U) is generated independently from D and we have the following equality in distribution: (UYT , YT ) d= (U, YT ), the resulting prediction regions will be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' The process of sharing samples between y ∈ Y is based upon this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' While described as resampling datasets ˜D(i) conditional on D in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, the non-stationarity tests we describe really sample permutations πi, conditionally on D, and then set ˜D(i) equal to Dπi := ((Cπi(1), Xπi(1), Yπi(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (Cπi(T ), Xπi(T ), Yπi(T )));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we use qΠ to denote the corresponding (to q) resampling distri- bution over permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' In the naive interval construction procedure which uses independent exogenous randomness, let Dy Π denote the (multi)set of permutations sampled for determining if y is in the acceptance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We claim that all these samples can be shared so that each y ∈ Y instead uses the entire set of shared permutations DΠ := � y∈Y Dy Π as opposed to just Dy Π: Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Let q be any conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' resampling distribution and let qΠ denote the correspond- ing resampling distribution over permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Furthermore define, for each y ∈ Y, a conditional distribution q′ Π,y over permutations which draws its samples conditionally independently from YT given (D[1:T −1], CT , XT ) and is defined by q′ Π,y(·|((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , YT ))) = qΠ (·|((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , y))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' To construct a prediction region, suppose that, for each y ∈ Y, a sampled (multi)set Dy Π = {π1,y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , πm,y} of permutations is generated by sampling each permutation conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from q′ Π,y(·|D)15, but the full (multi)set DΠ of m|Y| permutations is used to determine y’s membership in the acceptance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Specifically, defining Dy := ((C1, X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT , XT , y)), writing Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' y|Y|},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and defining Λ := {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 0)} ∪ ([m] × [|Y|]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we construct an acceptance region A from DΠ via the rule: y ∈ A ⇐⇒ � (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)∈Λ wq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y DΠ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )1[S((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) ≥ S(Dy)] > α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' (15) where wq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y DΠ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) := ˆf((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)∈Λ\\{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)} q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j(π˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j ◦ π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)∈Λ ˆf((Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) � (˜i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j′)∈Λ\\{(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)} q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j′ (π˜i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j′ ◦ π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' and we take π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y0 to simply be the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Then the resultant prediction region A controls miscoverage at the nominal rate under the proportionality hypothesis H∝ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Before giving a proof we make a brief computational/procedural note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' similar to equation (9) of Re- 15Since q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y does not depend on YT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we can indeed sample from this distribution during prediction region construction wherein YT is unobserved 41 mark 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' when q(·|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj) does not depend on (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' j) ∈ Λ for any y ∈ Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we have that wq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y DΠ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) = ˆf((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)∈Λ\\{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)} q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j(π˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j ◦ π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)∈Λ ˆf((Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) � (˜i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j′)∈Λ\\{(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)} q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j′ (π˜i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j′ ◦ π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) = ˆf((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )/q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ◦ π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)∈Λ ˆf((Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ◦ π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) = ˆf((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )/q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(id|(Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j′)∈Λ ˆf((Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (π−1 i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ |(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ )q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ (id|(Dy) πi′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj′ ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' where id denotes the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, computation of wq,y DΠ,i,yj((Dy)πi,yj ) across all (i, j) ∈ Λ becomes tractable in only O(m|Y|) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Consider a non-stationarity test in which we draw m|Y| samples and the resampling distribution ˘q we consider ignores YT , and instead draws these m|Y| samples in |Y| groups, the jth of which consists of m resamples Dπ1,yj , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Dπm,yj where the πi,yj are drawn conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' from q′ Π,yj (·|D) for each j ∈ [|Y|], and where Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , y|Y|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' It is important to note that ˘q is not a conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d resampling scheme and thus, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, the workaround discussed in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 must be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Defining Σ = {(0 ℓ) : ℓ ∈ [0 : m|Y|]} and letting i ∈ [0 : m] index with any given group and j ∈ � [|Y|] if i > 0 {0} if i = 0 index over groups, notice that wq,YT DΠ,i,yj(Dπi,yj ) = w˘q,Σ D,(i,j)( ˜D(i,j)), where D denotes the full set of m|Y| resamples and we double index resamples as D(i,j) = Dπi,yj for all (i, j) in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' This is because,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' letting q′ yj denote the induced (by q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj) distribution on each resample given by q′ yj(Dπ|D) = q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π|D) and looking at the rightmost terms in the numerator of wq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y DΠ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj((Dy)πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj )’s definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' we have that q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj(π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|Dπi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)∈Λ\\{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)} q′ Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j(π˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j ◦ π−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj|Dπi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) = q′ yj(D|Dπi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) � (˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)∈Λ\\{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)} q′ y˜j(D π˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='y˜j |Dπi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='yj ) = q′ yj(D| ˜D(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)) � (˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)∈Λ\\{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)} q′ y˜j( ˜D(˜i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='˜j)| ˜D(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j)) = � π∈Σ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='j) ˘q((( ˜D(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜D(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(1,|Y|), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜D(m,|Y|))π)−0| ˜D(i,j)), where Σ(i,j) is the single element subset of Σ containing solely the permutation on [0 : m|Y|] that swaps 0 and m(j − 1) + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Thus, the numerators of wq,YT DΠ,i,yj(Dπi,yj ) and w˘q,Σ D,(i,j)( ˜D(i,j)) are equal, and precisely the same argument shows that the denominators are also equal and thus wq,YT DΠ,i,yj(Dπi,yj ) = w˘q,Σ D,(i,j)( ˜D(i,j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Hence, the p-value � (i,j)∈Λ wq,y DΠ,i,yj(Dπi,yj )1[S(Dπi,yj ) ≥ S(D)], corresponding to line (15) is a valid p-value by the main result of Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Letting U denote the exogenous randomness used by this test, notice that the exogenous random variables Uy used by each y ∈ Y 42 in the prediction region construction of line (15) are all (deterministically) equal—since the only randomness in determining if y ∈ A is in the random sampling of permutations DΠ, which does not depend on the specific choice of y ∈ Y—and marginally identically distributed to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Since each of U and Uy is generated independently from YT , we have that (UYT , YT ) d= (U, YT ) and hence the corresponding prediction region A is valid, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' F Pseudocode for resampling distributions In this appendix section, we provide pseudocode for all resampling procedures described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Throughout this section, we use the notation Cat(p), for any p ∈ [0, 1]d with �d i=1 pi = 1, to denote the categorical distribution on [d] with probability of sampling i equal to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Non-stationarity testing in a C-stationary strongly non-reactive environ- ment We begin with the resampling procedures for non-stationarity testing in a C-stationary strongly non-reactive environment (Environment 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Algorithm 3: imitationπ Input: Data sequence D 1 Set ˜D to the empty list and set R ← [T] 2 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T do 3 Sample i ∼ Cat � � � PA(Xj| ˜D, Cj)1[j ∈ R] �T j′=1 PA(Xj′| ˜D, Cj′)1[j′ ∈ R] �T j=1 � � , if PA(·| ˜D)’s support intersects R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Append Zi to ˜D and set R ← R\\{i} Output: ˜D Algorithm 4: re-imitationπ Input: Data sequence D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' probability distribution PUt denoting the distribution of the tth exogenous random variable Ut generated by A 1 Set ˜D to the empty list and set R ← [T] 2 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T do 3 Sample ˜Ut ∼ PUt(·|∃s ∈ R : Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut)) if the conditioning event is non-empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Sample i ∼ Unif � {s ∈ R : Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut)} � 5 Append Zi to ˜D and set R ← R\\{i} Output: ˜D 43 Algorithm 5: cond-imitationπ Input: Data sequence D as well as the exogenous randomness U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , UT used to generate it 1 Set ˜D to the empty list and set R ← [T] 2 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T do 3 Sample i ∼ Unif � {s ∈ R : Xs = δt(Cs, ˜Ht−1, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ut)} � if the set is non-empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Append Zi to ˜D and set R ← R\\{i} Output: ˜D F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Non-stationarity testing in an MDP As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2, the datasets used in non-stationarity testing in an MDP are augmented with an additional action XT +1, and thus we may view these datasets as a list of T + 1 state action pairs: D = ((C1, X1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , (CT +1, XT +1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Under this framework, all resampling procedures in this section utilize the following function φ, which takes the state-action pair (c, x) to the set of indices which follow it: φ(c, x) = {t ∈ [2 : T + 1] : (Ct−1, Xt−1) = (c, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Additionally, using this view of the dataset D, we use Zt to denote the tth state-action pair (Ct, Xt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Zt denotes the tth state-action pair of the resampled dataset ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Using the function φ, we now present the corresponding uniformπ, imitationπ, re-imitationπ, and cond-imitationπ for an MDP under this setup of the dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Algorithm 6: uniformπ in an MDP Input: Data sequence D 1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] 2 for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T + 1 do 3 Sample i ∼ Unif � R ∩ φ( ˜Zt−1) � if the set is non-empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Append Zi to ˜D and set R ← R\\{i} Output: ˜D Algorithm 7: imitationπ in an MDP Input: Data sequence D 1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] 2 for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T + 1 do 3 Sample i ∼ Cat � � � PA(Xj| ˜D, Cj)1[j ∈ R ∩ φ( ˜Zt−1)] �T +1 j′=2 PA(Xj′| ˜D, Cj′)1[j′ ∈ R ∩ φ( ˜Zt−1)] �T +1 j=2 � � , if ∃j ∈ R ∩ φ( ˜Zt−1) such that PA(Xj| ˜D, Cj) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Append Zi to ˜D and set R ← R\\{i} Output: ˜D 44 Algorithm 8: re-imitationπ in an MDP Input: Data sequence D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' probability distribution PUt denoting the distribution of the tth exogenous random variable Ut generated by A 1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] 2 for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T + 1 do 3 Sample ˜Ut ∼ PUt(·|∃s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut)) if the conditioning event is non-empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Sample i ∼ Unif � {s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, ˜U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , ˜Ut)} � 5 Append Zi to ˜D and set R ← R\\{i} Output: ˜D Algorithm 9: cond-imitationπ in an MDP Input: Data sequence D as well as the exogenous randomness U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , UT +1 used to generate it 1 Set ˜D ← ((C1, X1)) and set R ← [1 : T + 1] 2 for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T + 1 do 3 Sample i ∼ Unif � {s ∈ R ∩ φ( ˜Zt−1) : Xs = δt(Cs, ˜Ht−1, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , Ut)} � if the set is non-empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Append Zi to ˜D and set R ← R\\{i} Output: ˜D F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Conditional independence testing We now give pseudocode for our resampling procedures for conditional independence testing discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We first show pseudocode for restricted-uniformπ resampling, which, although not used on its own for conditional independence testing, makes up the first stage of the restricted-uniformπ+imitationX resampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' We then go on to present the imitationX resampling procedure, which is also a key ingre- dient that is used in the uniformπ+imitationX and restricted-uniformπ+imitationX resampling procedures, which are presented subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Finally, we show pseudocode for combinedπ,X sampling, which combines the permutation and randomization of Xt’s into a single stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Algorithm 10: restricted-uniformπ Input: Data sequence D 1 Set ˜D to the empty list 2 Set Γ ← {π′ ∈ Π[T ] : g(Xπ′(t)) = g(Xt), ∀t ∈ [T]} 3 Sample π ∼ Unif(Γ) 4 ˜D ← (Zπ(t))T t=1 Output: ˜D G Supplementary simulation results G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 MCMC plots In this section, we present the power, Type-I error, coverage, and length plots for the unweighted MCMC randomization test and its inversion, for the inferential tasks discussed in Section 5, in Figures 9–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 45 Algorithm 11: imitationX Input: Data sequence D 1 Set ˜D to the empty list 2 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T do 3 Sample ˜Xt ∼ PA(·| ˜D, Ct, g(Xt)), if there exists x ∈ X with PA(x| ˜D, Ct, g(Xt)) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' terminate the sampling procedure 4 Append ˜Zt := (Ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' ˜Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Yt) to ˜D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Output: ˜D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Algorithm 12: uniformπ+imitationX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Input: Data sequence D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Sample D′ according to the uniformπ distribution applied to D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Sample ˜D according to the imitationX distribution (Algorithm 11) applied to D′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Output: ˜D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Algorithm 13: restricted-uniformπ+imitationX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Input: Data sequence D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1 Sample D′ according to the restricted-uniformπ distribution (Algorithm 10) applied to D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Sample ˜D according to the imitationX distribution (Algorithm 11) applied to D′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Output: ˜D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='Algorithm 14: combinedπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='X Input: Data sequence D 1 Set ˜D to the empty list and set R ← [T] 2 for t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' , T do 3 Sample i ∼ Cat � � � � � � x∈X:g(x)=g(Xj) PA(x| ˜D, Cj)1[j ∈ R] �T j′=1 � x′∈X:g(x′)=g(Xj′) PA(x′| ˜D, Cj′)1[j′ ∈ R] � � T j=1 � � � , if there exists j ∈ R such that � x∈X:g(x)=g(Xj) PA(x| ˜D, Cj) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 4 Set R ← R\\{i} 5 Sample ˜Xt ∼ PA(·| ˜D, Ci, g(Xi)), if there exists x ∈ X with PA(x| ˜D, Ci, g(Xi)) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' otherwise, terminate the sampling procedure 6 Append ˜Zi := (Ci, ˜Xt, Yi) to ˜D Output: ˜D G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 Computation times In this section, we plot the computation time curves (to compute p) for all resampling algorithms, environ- ments, adaptive assignment algorithms, and types of randomization test (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=', weighted MC or unweighted MCMC) discussed in Section 5 in Figures 15–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 46 Figure 9: Type-I error rate (leftmost) and power (second from left) of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T as well as power at fixed T = 100 and varying m (third from left) and fractional effective sample size plots at fixed m = 100 and varying T (rightmost) in a contextless stationary strongly non-reactive environment on data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 10: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='3 Auxiliary simulation results In this section we present all remaining auxiliary simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 23 displays the power plot at fixed T = 100 and varying m for the conditional independence test in the contextual stationary strongly non-reactive environment on data gathered via ϵ-greedy and LinUCB discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' On the other hand, Figure 24 illustrates that the phenomenon of shift in relative performance of the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline in comparison to both ϵ-greedy and LinUCB, described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='2 also occurs when the baseline is com- pared to a biased i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' adaptive assignment algorithm which selects actions at each timestep independently from 2Bern(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 47 Contextless C-stationary strongly non-reactive environment, non-stationanity test Type-I error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='100 Type-l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='6 0.' metadata={'source': 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m T +- uniform id baseline greedy cond-imitationn, MC greedy uniformn, MCMC greedycond-imitationn,MCMC +-greedyuniform,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB uniformn, MC greedyimitationr,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCB uniformn,MCMC +-greedyimitationr,MC UCBimitationr,MC greedy re-imitationr, MCMC .' metadata={'source': 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restricted-uniformn+imitationx, MCMC --greedyuniform,+imitationx,MC -greedy combinedn,x,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCBuniform,+imitationx,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB combinedn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='x, MCFigure 11: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy, LinUCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 12: Type-I error rate (leftmost) and power (second from right) of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T as well as power for fixed T = 100 and varying m (third from right) and fractional effective sample size at fixed m = 100 and varying T (rightmost) in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy and greedy Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 48 Contextual C-stationary strongly non-reactive environment, non-stationarity test Type-I error Power (fixed m) Power (fixed T) Fractional effective sample size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 25 50 75 100 50 75 100 102 103 104 25 75 100 T T m T +- uniform iid baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='. greedy uniformr, MC -greedy re-imitationn,MCMC --greedyuniformr,MC greedy imitation,MC *-greedy cond-imitation,MCMC greedyimitationr,MC *-greedyuniformn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='greedy uniformr,MCMC --greedyre-imitation,MC *-greedyimitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='. greedy imitationn, MCMC greedy cond-imitationn, MCFigure 13: Coverage and average length of confidence intervals for b0 using both weighted MC and unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 14: Coverage and average length of conformal prediction intervals for YT using both weighted MC and unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 49 Confidence interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='96 Average Length 6 Cove 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='92 eja 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='90 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB restricted-uniform,+imitationx,MCMC greedyuniformn+imitationx,MC 一-greedycombinedn,x,MCMC UCB uniformn+imitationx,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='. UCB combinedn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='x, MCContextless C-stationary strongly non-reactive environment, conformal prediction interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='00 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='96 abe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Average Length 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCB uniformn,MCMC --greedyimitationn,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCBimitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.MC greedy re-imitationr,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB imitationn,MCMC +-greedyre-imitationn,MCFigure 15: Computation times under the null (left) and alternative (right) distributions of randomization tests at fixed m = 100 and varying T in a contextual stationary strongly non-reactive environment on data gathered via ϵ-greedy and LinUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Note that the computation times for LinUCB imitationX are omitted as both Type-I error and power curves are simply generated i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Bern(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 16: Computation times under the null (left) and alternative (right) distributions of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless stationary strongly non- reactive environment on data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 50 Contextual stationary strongly non-reactive environment conditional independence test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' computation times 14 14 12 12 (seconds) AverageTime (seconds) 10 10 8 8 6 6 4 N 20 40 60 80 40 Q9 80 100 20 T T uniformiidbaseline E-greedyuniform+imitationx E-greedyimitationx(priorwork) LinUcB uniform+imitationxNull distribution Alternative distribution 5 2 20 40 Q9 80 100 20 40 09 80 100 T T +-uniformidbaseline UCB restricted-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='+imitationx,MC greedyuniformn+imitationx,MCMC greedyrestricted-uniform+imitationx,MC UCB uniformn+imitationx,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCB combineds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='x -greedy combinedn,x, MC greedy restricted-uniformn+imitationx,MCMC -*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCBrestricted-uniform+imitationx,MCMC greedyuniform,+imitationx,MC ε-greedy combinedn,x, MCMC UCB uniformn+imitationx, MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCB combinedn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 17: Computation times under the null (left) and alternative (right) distributions of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 18: Computation times under the null (left) and alternative (right) distributions of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy, LinUCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 51 Contextless C-stationary strongly non-reactive environment, non-stationarity test, average computation times Null distribution Alternative distribution QT 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 8 Average Time (seconds) 6 4 2 2 20 40 09 08 100 20 40 09 80 100 T T +- uniformiid baseline greedy cond-imitationn,M *-greedyuniformn,MCMC *-greedy cond-imitationn,MCMc greedyuniformn,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBuniform,MC --greedyimitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCBuniformnMCMC greedyimitationr,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB imitationn,MC greedy re-imitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBimitationn,MCMC greedy re-imitationn, MCContextual c-stationary strontly non-reactive environment, non-stationarity test, computation times Null distribution Alternative distribution 300 DCE 250 250 (seconds) 200 200 150 150 abe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 100 JaAY 100 50 [0] 0 20 40 60 80 100 20 40 60 80 100 T T +- uniform iid baseline greedy cond-imitationn,M *-greedyuniformr,MCMC greedycond-imitationn,MCMc greedyuniformn,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCB uniformn, MC greedyimitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.LinUCB uniformn,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='MCMC greedyimitation,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCB imitationn,MC E-greedyre-imitationn,McMc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='LinUCB imitationn,MCMC greedy re-imitationn, MCFigure 19: Computation times under the null (left) and alternative (right) distributions of both weighted MC and unweighted MCMC randomization tests at fixed m = 100 and varying T in a contextless C-stationary strongly non-reactive environment with data gathered via ϵ-greedy and greedy Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 20: Computation time of construction of confidence interval for b0 using both weighted MC and unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 52 MDP, non-stationarity test, computation times Null distribution Alternative distribution 14 14 12 12 (seconds) Average Time (seconds) 10 10 8 8 6 6 4 2 N 0 0 20 40 60 80 100 20 40 60 08 100 T T +- uniform iid baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='greedy uniformn, MC -greedy re-imitationn,MCMC --greedyuniformn,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.greedyimitationMC greedycond-imitation,MCMC greedyimitationr,MC *-greedyuniformn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.greedy uniformn,MCMC --greedyre-imitationn,MC -greedy imitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.greedy imitationn,MCMC greedycond-imitationn,MCConfidence interval, computation times 45 40 35 CE 25 20 15 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 0 20 40 60 80 100 T + uniformiid comparator UCB restricted-uniform,+imitationx,MC greedy uniform,+imitationx,MCMC greedy restricted-uniformn+imitationx,MC UCB uniformn+imitationx,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='. UCB combinedn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' greedy combinedn,x, MC greedy restricted-uniform,+imitationx,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='. UCB restricted-uniformn+imitationx, MCMC greedyuniformn+imitationx,MC greedy combinedr,x,MCMC UCB uniformn+imitationx, MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCB combinedn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='xFigure 21: Computation time of construction of conformal prediction interval for YT using both weighted MC and unweighted MCMC randomization tests with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 22: Computation time of construction of conformal prediction interval for YT using the MC randomization test with data gathered via ϵ-greedy, UCB, and the uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' 53 Contextless C-stationary strongly non-reactive environment, conformal prediction interval, computation times 70 Q9 50 40 20 10 20 40 09 08 100 T +-uniformiidbaseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBimitationn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='MC greedyimitationr,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBimitation,MCMC +-greedyimitationn,MC + -greedy uniformn,MC greedy re-imitationn, MCMC 一-greedyuniformn,MCMC -greedy re-imitationn,MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBuniformuMC greedycond-imitationn,MCMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBuniform,MCMC greedy cond-imitationn,MCContextual C-stationary strongly non-reactive environment, conformal prediction interval, shared samples, computation times m = 10 Mc samples m =1oo Mc samples 50 8 Average Time (seconds) 5 CE 20 2 10 0 20 40 60 80 100 20 40 60 80 100 T 人 uniform iid baseline greedy re-imitationr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='.UCBimitationn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='UCBuniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' greedy imitationr greedycond-imitation E-greedy uniformmFigure 23: Power of randomization tests at fixed T = 100 and varying m in a contextual stationary strongly non- reactive environment on data gathered via ϵ-greedy and LinUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' Figure 24: Power comparison of uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline versus biased i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content=' baseline that selects actions independently from 2Bern(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='1) − 1 54 Contextual stationary strongly non-reactive environment conditional independence test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE4T4oBgHgl3EQf-w4H/content/2301.05365v1.pdf'} +page_content='8 Average Power 0.' metadata={'source': 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b/R9AyT4oBgHgl3EQfVPdi/content/tmp_files/2301.00140v1.pdf.txt @@ -0,0 +1,1890 @@ +arXiv:2301.00140v1 [math.OA] 31 Dec 2022 +1 +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, +DIRICHLET SPACES AND DISCRETE GROUPS +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +Abstract. We study natural conditions on essentially discrete spectral triples (A, h, D) by +which the quantum differential da of a ∈ A belongs to the ideal generated by the unit length +ds = D−1. We also study upper and lower bounds on the singular values of the da’s and +apply the general framework to natural spectral triples of Dirichlet spaces and, in particular, +those on dual of discrete groups arising from negative definite functions. +1. Introduction +In Noncommutative Geometry [Co] a spectral triple (A, h, D) is made by a ∗-algebra of +bounded operators A ⊆ B(h) on a Hilbert space h on which a densely defined self-adjoint +(Dirac) operator D also acts in such a way that the commutators [D, a] ∈ B(h) are bounded +for all a ∈ A. The unit length or line element of (A, h, D) ([Co] Chapter 7), given by the +compact operator +ds = D−1, +is meant to encode the NC metric aspects of the structure. The volume element, given by +the spectral density operator dv = ρ(D) [CS3], where ρ(x) := N|D|(x)−1 is the reciprocal of +the counting function of |D|, represents the NC measure theoretic aspects. It can be written +as dv = σ(ds) where σ(x) := N|D|(x−1)−1 is the spectral density function of the compact +operator ds. Other fundamental infinitesimals (i.e. compact operators) are the quantum +differentials da := i[F, a] of elements a ∈ A, where (A, h, F) is a Fredholm module associated +to the spectral triple. In the classical case of the Riemannian geometry of a compact smooth +manifold, where A is the algebra of smooth functions, D is the Dirac operator and F is +the Hilbert transform, connections among the da’s, ds and their singular values, can be +established thanks to the tools of the pseudo-differential calculus. Even in this commutative +setting, however, a much finer analysis is needed when one only requires A to be an algebra +of Lipschitz or Sobolev functions, as in the case of the one dimensional fractals and quasi- +conformal manifolds (cf. works of A. Connes-D. Sullivan in [Co] and M. Hilsum in [Hil]). As +a rule, the higher smoothness of an element a ∈ A has to be read by the stronger compactness +of its quantum differential da which, in ultimate analysis, will lie in the ideal generated by +the unit length ds. +In this work we estimates the singular values of the da′s in terms of those of ds without +imposing any spectral growth condition on the Dirac operator D and only requiring minimal +and natural smoothness: A is a subalgebra either in the domain AD of the generator of the +automorphisms group α(a, t) := eitDae−itD, or it is lying in the intersection AD ∩ A|D| with +the domain A|D| of the generator of the automorphisms group β(a, t) := eit|D|ae−it|D| (in other +words, the commutators with D or |D| are bounded). In Quantum Mechanics α represents +the Heisenberg time evolution of observables in a system governed by the Dirac Hamiltonian +1This work has been supported by Laboratoire Ypatia des Sciences Mathématiques C.N.R.S. France - +Laboratorio Ypatia delle Scienze Matematiche I.N.D.A.M. Italy (LYSM). +Date: December 30, 2022. +2020 Mathematics Subject Classification. 58B34, 31C25, 46L57, 47A11. +1 + +2 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +D while, in Riemannian Geometry, β represents the geodesic flow, by the Egorov Theorem. +In order to deal with more examples, among which the forthcoming ones, we allow the Dirac +operator to have an infinite dimensional kernel, while it is assumed to have discrete spectrum +away from its kernel, i.e. to have compact inverse D−1 on the orthogonal ker(D)⊥ of its +kernel. +To the range of application of NCG, we add, with this work, Dirichlet spaces. These are C∗ +or von Neumann algebras, commutative or not, with a privileged quadratic form E satisfy- +ing a characteristic contraction property and named Dirichlet form, representing an energy +functional. This functional generalizes the Dirichlet integral of a Riemannian manifold and +allows to develop a kernel-free NC Potential Theory. In a Dirichlet space, the Dirac operator +is a 2 × 2 matrix +D = +� +0 +∂∗ +∂ +0 +� +defined by the derivation ∂, which represents the differential square root of the Dirichlet +energy form (cf. [CS1]) in the sense that +E[a] = ∥∂a∥2 +H. +Natural examples come from ground state representations of Hamiltonian in Quantum Me- +chanics, completely positive, Markovian semigroups converging to KMS equilibria in Quan- +tum Statistical Mechanics and generators of quantum Levy processes in Quantum Probability. +In the final part of the work we study, in particular, quantum differentials of Dirichlet spaces +associated to negative definite functions on countable, discrete groups. +A somewhat detailed account of the content of the work is as follows. +In Section 2, we +introduce essentially discrete spectral triples (A, h, D), a slight enlargement of the classical +notion which will be needed for applications to Dirichlet spaces. In the section, we consider +a canonical Fredholm operator F0 and a technical variant F of it, associated to any of these +triples, showing that they furnish equivalent Fredholm modules on A. The analysis is based +on old and new representations of the quantum differentials da in terms of the gradient +i[D, a] and the line element ds. In particular, it is shown that the da’s belong to the ideal +generated by +� +|ds| and that the singular values µ4k(da) are controlled by the singular values +µk(ds) with ratios proportional to the Lipschitz seminorms ∥[D, a]∥. Assuming that also +the commutators [|D|, a] are bounded, we prove that the da’s belong to the ideal generated +by ds and that the µ2k(da)’s are dominated by the µk(ds)’s. Their ratios are shown to be +asymptotically close to five times the sum of the seminorms ∥[D, a]∥, ∥[|D|, a]∥. In case of +discrete spectral triples the estimate is improved to control µk+d(da) where d := dim ker(D). +In Section 3, we introduce the canonical, essentially discrete spectral triple and Fredholm +module of a Dirichlet space (E, F) on a C∗-algebra with l.s.c. faithful trace (A, τ). The Dirac +operator is here the anti-diagonal matrix of the derivation ∂ and divergence ∂∗ canonically +associated to E, and A is the Lipschitz algebra AE made by elements of the Dirichlet algebra +B := F ∩ A which have bounded carré du champ (alias energy density). Elements of the +smooth sub-algebra A2,∞ +E +⊆ AE, which belong to the domain of the generator L of E on the +von Neumann algebra M := L∞(A, τ), are shown to have bounded commutators with |D|. +These results are consequences of the compactness on L2(A, τ) of the commutators with the +roots (I + L)γ when γ ∈ (0, 1/2) and their boundedness when γ = 1/2. +In Section 4 we prove lower bounds on the singular values of a quantum differential da when +the commutator [ +√ +I + L, a] is compact. +In this case, the singular values of the da’s are +controlled both above and below by those of the line element ds. The case of Dirichlet forms +whose generator are roots Lβ by β ∈ (0, 1) of L is especially considered. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS3 +In the final Section 5, we analyze Dirichlet forms constructed by negative type functions on +discrete groups. We introduce the class of slow negative type functions to which the entire +above analysis apply, showing also that it contains the length function of free groups and any +root of any proper negative type function on any discrete group. +2. Essentially discrete spectral triples and their Fredholm modules +To cover applications to Dirichlet spaces, we consider spectral triples of the following type: +Definition 2.1. (A, h, D) is said essentially discrete if σ(D) \ {0} is discrete. +Thus, if 0 ∈ σ(D), this value is allowed to be an eigenvalue with infinite degeneracy i.e., +denoting by P0 the orthogonal projection onto ker (D), we are assuming that (I − P0)D has +discrete spectrum as an operator on (I − P0)(h). +We shall adopt the convention by which D−1 and |D|−1 denote the operators which identically +vanish on ker (D) = P0h and are the usual functional calculi on (I − P0)(h). +With this +convention, both these operators are compact and we have the identities +DD−1 = D−1D = |D| |D|−1 = |D|−1|D| = I − P0, +D|D|−1 = sign(D) +where the last one is the sign operator corresponding to the sign function on R. The nonzero +part of the spectrum of |D| will be enumerated as +0 < λ1(|D|) ≤ λ2(|D|) · · · ≤ λn(|D|) ≤ λn+1(|D|) ≤ · · · +where each eigenvalue λn(|D|) is repeated according to its multiplicity. +2.1. Fredholm operators and quantum differentials. In the commutative situation +where D is the Dirac operator on the Clifford algebra of a closed, compact, Riemannian +manifold M, the sign operator Fcl := sign(D) is a 0-order ΨDO, since D is differential +operator and |D| is a ΨDO both of order 1. By the pseudo-differential calculus, the com- +mutators [Fcl, a] with smooth functions a ∈ C∞(M) are ΨDO of order −1 so that [Fcl, a]|D| +and |D|[Fcl, a] are bounded as 0-order ΨDO. In other words, the commutators [Fcl, a] with +smooth functions belong to the symmetric principal ideal (of compact operators) generated +by the line element ds = |D|−1. This property cannot be derived if the functions a are just +Lipschitz because in these cases pseudo-differential calculus does not apply. The purpose of +this section is to get similar estimates in the general noncommutative geometrical workframe +under rather mild assumptions on a ∈ B(h) of Lipschitz nature. +To consider the general situation, we associate to the Dirac operator, the following self-adjoint, +bounded operator +F := P0 + +D +√ +1 + D2 +which is in fact a Fredholm one, as it follows from +F 2 − I = P0 + (I − P0) +D2 +I + D2 − Ih = −(I − P0) +I +I + D2. +An alternative self-adjoint, Fredholm operator will be also considered: +F0 := P0 + D |D|−1. +It is an orthogonal symmetry which is equal to F up to a compact operator. More precisely +(2.1) +F 2 = I, +F − F0 = (I − P0)T |D|−2 + +4 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +where T := − D +|D| +D2 +√ +I + D2(|D| + +√ +I + D2) is a contraction. +The operators i[F0, a], i[F, a], both denoted by da, are the quantum differentials of a ∈ A +([Co] Chapter 4). Let us start with an observation: +Lemma 2.2. The following identity holds true +[P0, a] = P0 α0 |D|−1 + |D|−1β0 P0 +a ∈ A +for suitable operators α0, β0 ∈ B(h) with ∥α0∥ = ∥β0∥ = ∥[D, a]∥. +Proof. The stated representation follows from the identities +P0a − P0aP0 = P0a(I − P0) = P0aDD−1 = P0[a, D]D−1 = P0[a, D]|D|D−1|D|−1 +P0aP0 − aP0 = (I − P0)aP0 = D−1DaP0 = D−1[D, a]P0 = |D|−1|D|D−1[D, a]P0. +□ +Proposition 2.3. For a = a∗ ∈ A we have the bounds +(2.2) +− || [D, a] || (I + D2)−1/2 ≤ i +� +D +√ +1 + D2, a +� +≤ || [D, a] || (I + D2)−1/2 +(2.3) +− || [D, a] || |D|−1 ≤ i (I − P0) +� +D +√ +1 + D2, a +� +(I − P0) ≤ || [D, a] || |D|−1 +and +(2.4) +− ∥[D, a]∥ · |D|−1 ≤ i (I − P0) +� D +|D|, a +� +(I − P0) ≤ ∥[D, a]∥ · |D|−1. +Proof. Notice that the double inequality (2.3) is a straightforward consequence of (2.2), since +(I − P0) +1 +√ +1 + D2(I − P0) ≤ |D|−1. In order to prove the double inequality in (2.2), we use, +as usual (see [SWW] Proposition 1), the identity (in the strongly convergent sense) +D +√ +I + D2 = 1 +π +� +∞ +0 +t−1/2 +D +tI + I + D2dt + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS5 +from which we deduce +i +� +D +√ +1 + D2, a +� += 1 +π +� +∞ +0 +t−1/2 i +� +D +tI + I + D2, a +� +dt += 1 +π +� +∞ +0 +t−1/2 +� +I +tI + I + D2i[D, a] + I +� +I +tI + I + D2, a +� +D +� +dt += 1 +π +� +∞ +0 +t−1/2 +� +I +tI + I + D2i[D, a] − +I +tI + I + D2i[D2, a] +D +tI + I + D2 +� +dt += 1 +π +� +∞ +0 +t−1/2� +I +tI + I + D2i[D, a] − +D +tI + I + D2i[D, a] +D +tI + I + D2 +− +I +tI + I + D2i[D, a] +D2 +tI + I + D2 +� +dt += 1 +π +� +∞ +0 +t−1/2� +I +tI + I + D2i[D, a] +� +I − +D2 +tI + I + D2 +� +− +D +tI + I + D2i[D, a] +D +tI + I + D2 +� +dt += 1 +π +� +∞ +0 +t−1/2� +tI + I +tI + I + D2i[D, a] +I +tI + I + D2 +− +D +tI + I + D2i[D, a] +D +tI + I + D2 +� +dt . +Noticing now that i[D, a] is self-adjoint, so that −|| [D, a] || I ≤ i[D, a] ≤ || [D, a] || I, the +above identity thus provides +i +� +D +√ +1 + D2, a +� +≤ || [D, a] || 1 +π +� +∞ +0 +t−1/2� +tI + I +(tI + I + D2)2 + +D2 +(tI + I + D2)2 +� +dt += || [D, a] || 1 +π +� +∞ +0 +t−1/2 +I +tI + I + D2 dt += || [D, a] || (I + D2)−1/2 +and, similarly, +i +� +D +√ +1 + D2, a +� +≥ −|| [D, a] || 1 +π +� +∞ +0 +t−1/2� +tI + I +(tI + I + D2)2 + +D2 +(tI + I + D2)2 +� +dt += −|| [D, a] || 1 +π +� +∞ +0 +t−1/2 +I +tI + I + D2 dt += −|| [D, a] || (I + D2)−1/2 . +The double inequality (2.4) can be proved the same way, with obvious adaptations, or derived +from Proposition 1 in [SWW] where the same inequality is stated in case D is invertible. In +fact, on one hand +i (I−P0) +� D +|D|, a +� +(I−P0) = i +� +(I−P0) D +|D| (I−P0) , (I−P0)a(I−P0) +� += i +� D +|D|, (I−P0)a(I−P0) +� +and, since ∥ +� +D, (I −P0)a(I −P0) +� +∥ = ∥[D, a]∥, (D, (I −P0)h, (I −P0)A(I −P0)) is a spectral +triple where D is invertible on (I − P0)h. +□ + +6 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +Using the bounds above, we show that the quantum differentials of elements of A belong to +the symmetric principal ideal in B(h) generated by +� +|ds| = |D|−1/2. +Proposition 2.4. For any fixed a = a∗ ∈ A +i) there exist bounded operators α, β, γ ∈ B(h) such that +i[F, a] = α|D|−1 + |D|−1β + |D|−1/2γ |D|−1/2, +with ||α|| ≤ 2 || [D, a] ||, ||β|| ≤ 2 || [D, a] ||, ||γ|| ≤ || [D, a] ||; +ii) a similar representation holds true with F0 in place of F. +Proof. i) For [P0, a], see Lemma 2.2 above. Since DP0 = 0 we have +P0[(1 + D2)−1/2D, a] = −P0aD(1 + D2)−1/2(I − P0) = P0[D, a](1 + D2)−1/2(I − P0) += P0[D, a] +|D| +√ +1 + D2|D|−1 = P0[D, a] +|D| +√ +1 + D2|D|−1(1 − P0). +Similarly, +[(1 + D2)−1/2D, a]P0 = (1 − P0)|D|−1 +|D| +√ +1 + D2 [D, a]P0. +As for (I −P0)[(1+D2)−1/2D, a](I −P0) = (I −P0)[(1+D2)−1/2D, (I −P0)a(I −P0)](I −P0), +one invokes (2.3) of Proposition 2.3, which is equivalent to the assertion +−|| [D, a] || |D|−1 ≤ i +� +D +√ +I + D2, (I − P0)a(I − P0) +� +≤ −|| [D, a] || |D|−1 . +ii) follows from (2.4) in Proposition 2.3 and the same proof as above. +□ +As first consequence of the above representation, we have +Corollary 2.5. If (A, h, D) is an essentially discrete spectral triple, then (A, h, F0) and +(A, h, F) are, essentially unitary equivalent, Fredholm modules. +Proof. Follows from the identity (2.1), Lemma 2.2 and Proposition 2.3. +□ +A second consequence concerns a bound on the singular values of the quantum differentials. +Corollary 2.6. If (A, h, D) is an essentially discrete spectral triple, the singular values of +quantum differentials are controlled by the Lipschitz seminorm and the singular values of D−1 +µ4k(i[F0, a]) ≤ 5 ∥ [D, a] ∥ µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 +a = a∗ ∈ A, +k ≥ 0, +µ4k(i[F, a]) ≤ 5 ∥ [D, a] ∥ µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 +a = a∗ ∈ A, +k ≥ 0. +In terms of line element and quantum differentials, we proved the bounds +µ4k(da) ≤ 5 ∥ [D, a] ∥ µk(ds) +a = a∗ ∈ A, +k ≥ 0. +Proof. Applying Proposition 2.4 and the rules of singular values [Co] Chapter 4 Appendix C, +we have +µ4k(i[F, a]) ≤ µk(α|D|−1) + µk(|D|−1β) + µ2k(|D|−1/2γ |D|−1/2) +≤ ∥α∥µk(|D|−1) + ∥β∥µk(|D|−1) + ∥γ∥µk(|D|−1/2)2 +≤ 5 ∥ [D, a] , µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 . +and the same for µ4k(i[F0, a]). +□ + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS7 +2.2. Improving the estimates for the singular values of quantum differentials. Here +we consider a natural condition by which the quantum differentials da of elements a ∈ A +belong to the principal ideal in K(h) generated by the line element ds. The quantum differ- +entiation operator d can then be considered as a derivation from A to the ideal ID ⊆ K(h) +generated by ds, seen as a A-bimodule over the norm closure A := A ⊆ B(h). +Lemma 2.7. Let a ∈ A be such that the commutator [ |D|, a ] is bounded. Then one has +[P0, a](I − P0) = P0[a, D]F0 |D|−1 +� D +|D|, a +� +(I − P0) = −D|D|−1[ |D|, a] |D|−1 + [D, a] |D|−1 +� +D +√ +I + D2, a +� +(I − P0) = − +D +√ +I + D2 +�√ +I + D2, a +� +|D| +√ +I + D2|D|−1 + [D, a] +|D| +√ +I + D2|D|−1 +(2.5) +and +[P0, a]P0 = −|D|−1F0[D, a]P0 +� D +|D|, a +� +P0 = |D|−1[D, a]P0 +� +D +√ +I + D2, a +� +P0 = |D|−1 +D +√ +I + D2[D, a]P0 . +(2.6) +Proof. Let us compute successively : +[P0, a](I − P0) = P0a(I − P0) = P0aDD−1 = P0[a, D]D−1 = P0[a, D]F0|D|−1 +� D +|D|, a +� +(I − P0) = D[ |D|−1, a](I − P0) + [D, a] |D|−1 += D(I − P0)[ |D|−1, a](I − P0) + [D, a] |D|−1 += −D|D|−1[ |D|, a] |D|−1 + [D, a] |D|−1 +� +D +√ +I + D2, a +� +(I − P0) = D +� +I +√ +I + D2, a +� +(I − P0) + [D, a] I − P0 +√ +I + D2 += D(I − P0) +� +|D|−1, a +� +(I − P0) + [D, a] |D|−1 += −D +I +√ +I + D2 +� √ +I + D2, a +� I − P0 +√ +I + D2 + [D, a] I − P0 +√ +I + D2 += −D +I +√ +I + D2 +� √ +I + D2, a +� +ID| +√ +I + D2 |D|−1 + [D, a] +|D| +√ +I + D2 |D|−1 +[P0, a]P0 = −(I − P0)aP0 = −D−1[D, a]P0 = −|D|−1F0[D, a]P0 +� D +|D|, a +� +P0 = D +|D|aP0 = |D|−1[D, a]P0 +� +D +√ +I + D2, a +� +P0 = +D +√ +I + D2aP0 += +I − P0 +√ +I + D2 +� +D, a +� +P0 += |D|−1 +|D| +√ +I + D2 [D, a] P0 . + +8 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +□ +Proposition 2.8. Let a ∈ A be such that the commutator [ |D|, a ] is bounded. Then there +exist bounded operators α0 and β0 such that +i[F0, a] = α0 |D|−1 + |D|−1 β0 +with ∥α0∥ ≤ ∥ [D, a] ∥, ∥β0∥ ≤ ∥ [D, a] ∥+∥ [ |D|, a ] ∥ and β0P0 = 0. The quantum differential +da = i[F0, a] thus belongs to the symmetric ideal ID ⊆ B(h) generated by the line element ds. +Proof. This is a straightforward consequence of the previous Lemma 2.7. +□ +The similar result for the commutator [F, a] needs a preliminary lemma: +Lemma 2.9. Let a ∈ A be such that the commutator [|D|, a] is bounded. Then +i) the following bound holds true ∥[ +√ +1 + D2, a] ∥ ≤ ∥[|D|, a] ∥ + 2∥a∥. +ii) one has +∥(I − P0) +�√ +1 + D2, a +� +(I − P0)∥ ≤ C1(λ1)∥ +� +|D|, a +� +∥. +where C1(λ1) = +� +1 + λ−2 +1 +and λ1 := λ1(|D|) > 0 is the first nonzero eigenvalue of |D|. +Proof. The estimate in i) follows from the identity +√ +1 + D2 = |D| + +I +√ +1 + D2 + |D| and the +bound ∥( +√ +1 + D2 + |D|)−1∥ ≤ 1. The same identity also implies +(I − P0 +� +[ +√ +1 + D2, a +� +(I − P0) =(I − P0) +� +|D|, a +� +(I − P0) +− +I − P0 +√ +1 + D2 + |D| +� √ +1 + D2, a +� +I − P0 +√ +1 + D2 + |D| +− +I − P0 +√ +1 + D2 + |D| +� +|D|, a +� +I − P0 +√ +1 + D2 + |D| +and the inequality +�� (I − P0)[ +√ +I + D2, a +� +(I − P0) +�� ≤ +�� [ |D|, a +��� + ( +� +1 + λ2 +1 + λ1)−2�� (I − P0)[ |D|, a +� +(I − P0) +�� ++( +� +1 + λ2 +1 + λ1)−2�� (I − P0)[ +√ +I + D2, a +� +(I − P0) +�� +which in turn provides +� +1 − ( +� +1 + λ2 +1 + λ1)−2��� (I − P0)[ +√ +I + D2, a +� +(I − P0) +�� ≤ +� +1 + ( +� +1 + λ2 +1 + λ1)−2� +∥[ |D|, a +� +∥ +and the result with +C1(λ1) = 1 + ( +� +1 + λ2 +1 + λ1)−2 +1 − ( +� +1 + λ2 +1 + λ1)−2 = 1 + ( +� +1 + λ2 +1 − λ1)2 +1 − ( +� +1 + λ2 +1 − λ1)2 = 1 + λ2 +1 − λ1 +� +1 + λ2 +1 +λ1 +� +1 + λ2 +1 − λ2 +1 += +1 +λ1 +� +1 + λ2 +1 − λ2 +1 +− 1 = +� +1 + λ2 +1 + λ1 +λ1 +− 1 = +� +1 + λ−2 +1 . +□ +Remark 2.10. One can slightly improve estimate i) above by replacing ||a|| by the norm of +a in the quotient space A/A ∩ {D}′. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS9 +Proposition 2.11. Let a ∈ A be such that the commutator [ |D|, a ] is bounded. Then there +exist bounded operators α1, β1 ∈ B(h) such that +i[F, a] = α1|D|−1 + |D|−1β1 +with ∥α1∥ ≤ 2∥ [D, a] ∥, ∥β1∥ ≤ 3∥ [D, a] ∥+C1(λ1)|| [ |D|, a] ∥ (or, at choice, ∥β1∥ ≤ 3∥ [D, a ]∥+ +∥ [ |D|, a] ∥ + 2||a||) and β1 = β1 P0 and the quantum differential da = i[F, a] belongs to the +symmetric ideal ID ⊆ B(h) generated by the line element ds. +Proof. A straightforward consequence of Lemmas 2.7 and 2.9. +□ +Corollary 2.12. Let a ∈ A be such that the commutator [ |D|, a ] is bounded, then singular +values of the quantum differentials are controlled by +µ2k(i[F0, a]) ≤ (||α0|| + ||β0||) µk(|D|−1) = (||α0|| + ||β0||) ∥ λk+1(|D|)−1 +µ2k(i[F, a]) ≤ (||α1|| + ||β1||) µk(|D|−1) = (||α1|| + ||β1||) ∥ λk+1(|D|)−1 +with α0, β0, α2 and β1 provided by Propositions 2.8 and 2.11. In terms of line element and +quantum differentials, we have +µ2k(da) ≤ const.µk(ds) +a = a∗ ∈ A, +k ≥ 0. +Proof. Apply Propositions 2.8 and 2.11 along the lines of the proof of Corollary 2.6. +□ +From this proposition, in case of discrete spectrum, we deduce the following : +Corollary 2.13. Let a ∈ A, a = a∗ such that commutator [ |D|, a] is bounded and suppose +that the kernel of D is finite dimensional. Then one has the estimates +µk+d(i[F0, a]) ≤ ||α0|| µk(|D|−1) = ||α0|| λk+1(|D|)−1 +µk+d(i[F, a]) ≤ ||α1|| µk(|D|−1) = ||α1|| λk+1(|D|)−1 . +with α0 and α1 are provided by Propositions 2.8 and 2.11 respectively and d = dim(ker(D)). +In particular, ||α0|| ≤ || [D, a] || and ||α1|| ≤ 2|| [D, a] ||. In terms of line element and quantum +differentials, we have +µk+d(da) ≤ const.µk(ds) +a = a∗ ∈ A, +k ≥ 0. +2.3. The same estimates from an asymptotic point of view. Here we prove under the +double Lipschitz assumptions above, estimates which are asymptotically a bit more precise. +Proposition 2.14. Let a ∈ A be such that the commutator [ |D|, a ] is bounded. Then there +exist bounded operators α3, β3, γ3 ∈ B(h) such that +i[F, a] = α3|D|−1 + |D|−1β3 + |D|−1γ3|D|−1 +with ∥α3∥ ≤ 2∥ [D, a] ∥, ∥β3∥ ≤ 3∥ [D, a] ∥ + || [ |D|, a] ∥ and γ3 bounded. +Proof. According to Lemma 2.7, we have +[P0, a] = σ0|D|−1 + |D|−1τ0 with ||σ0|| ≤ || [D, a] || ||τ0|| ≤ || [D, a] || +� +D +√ +I + D2, a +� +P0 = |D|−1τ1 with ||τ1|| ≤ || [D, a] || +� +D +√ +I + D2, a +�� +I − P0 +� += σ1|D|−1 − +D +√ +I + D2[ +√ +I + D2, a] +D +√ +I + D2 with ||σ1|| ≤ || [ |D|, a] ||. + +10 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +We compute +[ +√ +I + D2, a] = [ |D|, a] + +� +I +√ +I + D2 + |D| +, a +� += [ |D|, a] − +I +√ +I + D2 + |D| +� √ +I + D2 + |D|, a +� +I +√ +I + D2 + |D| +and notice that +� √ +I + D2+|D|, a +� +is a bounded operator. Summing up, we get the result. +□ +We need a Lemma which slightly ameliorates a result due to K. Fan ([GK] Ch. II par. 5 +Theorem 2.3). For sake of completeness we provide a detailed proof. +Lemma 2.15. Let T and σ be two compact operators. Then there exist an integer d1 and +two sequences (εk)k≥0 and (ε′ +k)k≥0 such that limk→∞ εk = limk→∞ ε′ +k = 0 and +(1 − ε′ +k)µk+d1(T) ≤ µk +� +T(I + σ) +� +≤ (1 + εk)µk(T) . +Proof. Let us recall ([Connes chapter 4 section 2]) that +µk(T) = inf ||P ⊥T|| = inf ||T Q⊥|| +where P or Q runs in the set of orthogonal projections with rank less than k, and that +the infimum is indeed a minimum, reached when P (resp. Q) is the orthogonal projection +corresponding to the k first larger eigenvalues of |T ∗| (resp. |T|). +Let Pk (resp. Qk) be the orthogonal projection corresponding the k first eigenvalues of |T ∗| +(resp. |T|) : we have µk(T) = ||P ⊥ +k T|| and PkT = TQk, hence P ⊥ +k T = TQ⊥ +k = P ⊥ +k TQ⊥ +k . +Notice that, as k → ∞, the Qk tend increasingly toward I − q0, so that the Q⊥ +k tend to +q0, where q0 is the orthogonal projection on the kernel of T. Notice that, as σ is compact, +limk→∞ ||Q⊥ +k (I − q0)σ|| = 0. Compute now +µk +� +T(I + σ) +� +≤ ||P ⊥ +k T(I + σ)|| += ||P ⊥ +k TQ⊥ +k (I − q0)(I + σ)|| +≤ ||P ⊥ +k T|| +� +1 + ||Q⊥ +k (I − q0)σ||) +which provides the right inequality, with εk = ||Q⊥ +k (I − q0)σ||. +Notice now that there exists a compact operator τ such that (I + σ)(I + τ) = I − p1, where +p1 is the orthogonal projection on ker(I + σ∗) = Im(I + σ)⊥. This is a finite rank projection, +with rank d1. Applying the inequality just proved above, we can write +µk+d1(T) = µk+d1 +� +T(I − p1) + Tp1 +� +≤ µk +� +T(I − p1) +� ++ µd1(Tp1) = µk +� +T(I − p1) +� += µk +� +T(I + σ)(I + τ) +� +≤ µk +� +T(I + σ) +� +(1 + �εk) +with limk→∞ �εk = 0. This ends the proof. +□ +Corollary 2.16. Let a ∈ A be such that the commutator [ |D|, a ] is bounded. Then one has +µ2k +� +i[F, a] +� +≤ +� +5|| [D, a] || + || |D|, a] || +� +(1 + εk) λk+1(|D|)−1 with lim +k→∞ εk = 0 . + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS11 +3. Spectral triples of Dirichlet spaces +In this section we construct the spectral triple of a Dirichlet space on a C∗-algebra with trace +and we apply the previous results to study the associated Fredholm module. +3.1. Dirichlet forms and their tangent bimodules. For the definition and properties of +Dirichlet forms on trace C∗-algebras we refer to [AHK], [C1], [C2], [CS1], [DL], [S]. We list +below their main properties we need and fix notations for the rest of the paper. +In the following, (A, τ) is a C∗-algebra equipped with a faithful, densely defined, lower semi- +continuous trace and M := πτ(A)′′ ⊆ B(L2(A, τ)) is the corresponding von Neumann algebra +acting on the Hilbert space of the GNS representation πτ : A → B(L2(A, τ)). +We consider a completely Dirichlet form (E, F) on L2(A, τ) and its densely defined, positive, +self-adjoint generator (L, D(L)) in such a way that E is the closure of the quadratic form +D(L) ∋ ξ → (ξ|Lξ) and one has F = D(L1/2) and E[ξ] = ∥L1/2ξ∥2 for ξ ∈ F. +Since now on, we shall assume that (L, D(L)) has discrete spectrum away from zero. +Among the characteristic properties of a Dirichlet form, we recall that +i) the semigroup {e−tL : t > 0} maps L2(A, τ) ∩ M into itself and extends to a σ-weakly +continuous, completely positive contraction semigroup of M, still denoted by the same symbol; +ii) the resolvent {(I + tL)−1 : t > 0} maps L2(A, τ) ∩ M into itself and extends to a σ-weakly +continuous, completely positive contraction resolvent of M, still denoted by the same symbol. +The generator of the semigroup on M, denoted by (L, DM(L)), has a domain +DM(L) := {x ∈ M : L(x) := lim +t↓0 (x − e−tLx)/t +exists σ − weakly in M} +which, for any t > 0, coincides with (I + tL)−1(M). +The Dirichlet algebra B := F ∩ A is an involutive subalgebra of A. We assume that (E, F) +is regular in the sense that B is dense both in A and L2(A, τ), in their respective topologies, +and that it is a form core. +3.2. Tangent bimodule of a Dirichlet space and carré du champ. Let us recall the +main results of [CS1]: to any regular Dirichlet form (E, F) is associated a symmetric A- +bimodule (H, J ) together with a symmetric derivation ∂ : B → H, i.e. a linear map satisfying +∂(a∗) = J (∂a) +a, b ∈ B, +and the Leibnitz rule +(3.1) +∂(ab) = a∂b + (∂a)b +a, b ∈ B, +which is closable as a densely defined operator from L2(A, τ) into H, and such that +E[a] = ∥∂a∥2 +H +a ∈ B . +The carré du champ or energy density of a ∈ B is the following positive linear form Γ[a] ∈ A∗ ++ +⟨Γ[a], b⟩ = (∂a|(∂a)b)H +b ∈ A . +A useful approximation of the carré du champ (cf. [CS1]) is the following one: +Lemma 3.1. i) For any ε > 0, Lε := +L +1 + εL generates a bounded Dirichlet form on L2(A, τ). +ii) Setting Γε[a] := 1 +2 +� +a∗Lε(a) + Le(a)∗a − Lε(a∗a) +� +∈ M ∩ L1(A, τ) for a ∈ B, one has +(3.2) +⟨Γ[a], b⟩ = lim +ε↓0 τ +� +Γε[a] b +� +b ∈ A. + +12 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +Remark 3.2. i) A regular Dirichlet form provides the C∗-algebra of a potential theoretic +structure which generalizes the classical Dirichlet integral on a Riemannian manifold; +ii) the derivation (∂, B) is closable and the domain of its closure coincides with the form +domain F. When no confusion can arise, we shall use the same notation ∂ for the closure; +iii) there is not, in general, a formula for the adjoint divergence operator ∂∗ from H to L2(A, τ), +except in case where there exists a subalgebra B0 of A contained in the domain of L, for which +∂∗(∂(a)b) = 1 +2 +� +L(ab) + L(a)b − aL(b) +� +, a, b ∈ B0. +In the classical case of the Dirichlet integral, the derivation coincides with the gradient oper- +ator and its adjoint with the divergence operator. An example of computation of derivation +and divergence when any such subalgebras B0 trivialize to the multiples of 1A, is given on +fractals in [CGIS]. +3.3. The Lipschitz algebra of a Dirichlet spaces. Here we isolate a subalgebra of the +Dirichlet algebra B which play the role of algebra of Lipschitz functions on a Riemannian +manifold. +Lemma 3.3. 1. For a ∈ B, the following conditions are equivalent: +i) the carré du champ Γ[a] is absolutely continuous with respect to the trace τ and its Radon- +Nikodym derivative dΓ[a] +dτ +is bounded (which we write shortly Γ[a] ∈ M) +(∂a|(∂a)b)H = τ(bΓ[a]) +b ∈ A; +ii) there exists a constant Ca ≥ 0 such that +|⟨Γ[a], b⟩| ≤ Caτ(|b|) +b ∈ B; +iii) the vector ∂a ∈ H is right-τ-bounded, i.e. there exists a constant Ca ≥ 0 such that +∥∂(a)b∥2 +H ≤ Caτ(b∗b), +b ∈ B. +2. The set AE ⊆ B whose elements and their adjoint satisfy the conditions above is a ∗- +subalgebra of B. +Proof. 1. is straightforward and 2. is a consequence of the symmetry and the Leibnitz rule +of the derivation (3.1). +□ +Definition 3.4. AE will be called the Lipschitz algebra of the Dirichlet space (E, F). For +a ∈ AE, we shall denote R(a) : L2(A, τ) → H the bounded operator characterized by +R(a) : L2(A, τ) → H +R(a)b := (∂a)b +b ∈ B . +Remark 3.5. Notice that due to the Leibnitz rule for ∂, one has, for b ∈ B +R(a)b = ∂(a)b = ∂(ab) − a∂b = [∂, a]b +so that a belongs to AE if and only if it has bounded commutator with the derivation ∂. +3.4. The smooth subalgebra. We show that the Lipschitz algebra AE contains a subalgebra +of elements in the operator domain of the generator. +Proposition 3.6. Let DM(L) be the domain of the generator on the von Neumann algebra. +Then the space +A2,∞ +E +:= AE ∩ DM(L) +is a ∗-subalgebra of the Lipschitz algebra AE. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS13 +Proof. Observe first that for t > 0, since (I + tL)−1 is a *-weakly continuous contraction of +M and E is symmetric with respect to τ, (I + tL)−1 will be also a contraction of the predual +L1(A, τ) = M∗. Notice then that B2 ⊂ L1(A, τ) is dense in L1(A, τ): in fact if x ∈ M is +orthogonal to B2, one has 0 = τ(bax) = (a∗|xb)L2(A,τ) for all a, b ∈ B so that x = 0. Hence +B ∩ L1(A, τ) is dense in L1(A, τ). Consider now a ∈ A2,∞ +E +. According to [CS], for b ∈ B, one +has +2⟨Γ[a], b⟩ = (L1/2(a)|L1/2(ab))L2(A,τ) + (L1/2(ab∗)|L1/2a)L2(A,τ) − (L1/2(a∗a)|L1/2b)L2(A,τ) +so that, if Γ[a] ∈ M and L(a) ∈ M, there exists a constant C such that +� +L1/2(a∗a)|L1/2b)L2(A,τ) +�� ≤ C τ(|b|) , b ∈ B ∩ L1(A, τ). +In particular, for any t > 0 and b ∈ B ∩ L1(A, τ), b ≥ 0 : +⟨L( +I +I + tL(a∗a)), b⟩L2(A,τ) = ⟨L1/2(a∗a), L1/2( +I +I + tL(b))⟩L2(A,τ) +≤ C τ( +I +I + tL(b)) ≤ Cτ(b) . +This estimate implies that +��L((I + tL)−1(a∗a)) +�� +M ≤ C is bounded uniformly on t > 0. As +lim +t↓0 (I +tL)−1(a∗a) = a∗a, σ-weakly in M, and L is a σ-weakly closed operator on M, we have +proved a∗a ∈ DM(L). +□ +3.5. Examples by proper, conditionally negative type functions on discrete groups. +Let G be a discrete group with the Haagerup property and ℓ a proper, conditionally negative +type function on G. The operator L of multiplication by ℓ in l2(G) = L2(C∗ +red(G), τ) is the +generator of the Dirichlet form +E : l2(G) → [0, +∞] +E[a] = +� +g∈G +|a(g)|2. +(τ being the canonical trace on the reduced C∗-algebra of G). +Let λ be the left regular representation of A = C∗ +red(G) and define A∞ as the algebra of +elements a = � +g∈G a(g)λ(g) in C∗ +red(G) whose sequence of Fourier coefficients has finite +support ({g ∈ G , a(g) ̸= 0} is finite). It is known that there exists a unitary representation +π of G in on a Hilbert space h and a 1-cocycle +c : G → h. +c(gg′) = c(g) + π(g)c(g′) +such that the conditionally negative type definite function ℓ can be represented as +⟨c(g′), c(g)⟩h = 1 +2 +� +ℓ(g) + ℓ(g′) − ℓ(g′−1g) +� +, +ℓ(g) = ∥c(g)∥2 , +g, g′ ∈ G. +The tangent bimodule is then H = h ⊗ l2(G) where G acts on the left by the diagonal +representation π ⊗λ and acts on the right by 1h ⊗ρ where ρ is the right action of G on ℓ2(G). +For a ∈ A∞, the derivation representing E is given by +∂a = +� +G +a(g)c(g) ⊗ δg , a ∈ A∞ + +14 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +and the carré du champ is indeed an element of A∞ +Γ[a] = +� +g1,g2∈G +a(g2) a(g1)⟨c(g2), c(g1)⟩hλ(g−1 +2 g1) += 1 +2 +� +g1,g2∈G +a(g2) a(g1) +� +ℓ(g2) + ℓ(g1) − ℓ(g−1 +2 g1) +� +λ(g−1 +2 g1). +So that A∞ ⊂ A2,∞ +L +⊂ AL, which implies that both AL and A2,∞ +L +are dense subalgebras of A. +Remark 3.7. Notice that Γ[λ(g)] = ℓ(g) Iℓ∞(G) is an invertible element of A∞ whenever +ℓ(g) ̸= 0, i.e. for elements of G outside a finite subset. +As a consequence, as < b, Γ[a]b >= ||R(a)b||2, one has +(3.3) +R(λ(g))∗R(λ(g)) = ℓ(g) Iℓ∞(G) +3.6. The spectral triple of a Dirichlet space. From now on and till the end of this paper, +we assume the generator L of the Dirichlet form E to have discrete spectrum away from its +kernel. Let us consider the triple +(L2(A, τ) ⊕ H , AE , D) +where the Dirichlet algebra AE, as a subalgebra of A, acts on L2(A, τ) ⊕ H by the diagonal +action of A on the left, both on L2(A, τ) and H and the Dirac operator is defined as +D = +� +0 +∂∗ +∂ +0 +� +. +Theorem 3.8. The triple (L2(A, τ)⊕H , AE , D) is an essentially discrete spectral triple. In +particular +i) the commutator of the derivation ∂ with the actions of AE on L2(A, τ) and H is given by +[∂, a ] = R(a) +for all +a ∈ AE +with +R(a)b = (∂a)b +for all +b ∈ B; +ii) for a ∈ AE we have ∥[D, a]∥ = max(∥R(a)∥, ∥R(a∗)∥) and +� +D, a +� += +� +0 +[∂∗, a] +[∂, a] +0 +� += +� +0 +−R(a∗)∗ +R(a) +0 +� +; +iii) in the polar decomposition ∂ = u L1/2 of the derivation ∂, the partial isometry u : +L2(A, τ) → H is such that u∗u = IL2(Aτ) − p0 and uu∗ = IH − q0, where p0 and q0 are +the orthogonal projections onto ker(∂) = ker(L) and ker(∂∗) = Im(∂)⊥, respectively; +iv) one has D2 = +� +L +0 +0 +uLu∗ +� +and |D| = +� +L1/2 +0 +0 +uL1/2u∗ +� += +� +u∗∂ +0 +0 +∂u∗ +� +; +v) if we enumerate λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · the nonzero eigenvalues of L , the corresponding +enumeration for |D| is +√ +λ1 ≤ +√ +λ1 ≤ +√ +λ2 ≤ +√ +λ2 ≤ · · · ≤ +√ +λn ≤ +√ +λn ≤ · · · i.e. λn(|D|) = +λ1/2 +[(n+1)/2] and µn(|D|−1) = λ−1/2 +[n/2]+1 ([r] being the integer part of a real r); +vi) the projection onto the kernel of D is P0 = +� +p0 +0 +0 +q0 +� +. +Proof. Straightforward by Lemma 3.3. +□ +On compact quantum groups, spectral triples of the above type has been constructed in +[CFK] Theorem 8.4, staring from the Dirichlet form of GNS-symmetric noncommutative +Levy processes. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS15 +Remark 3.9. In all interesting examples, ker(∂∗) is infinite dimensional and then, even if L +has a finite dimensional kernel (which often occurs), P0 may have, in general, infinite rank. +3.7. The Fredholm modules of a Dirichlet space. According to subsection 2.1 and with +the notations of Theorem 3.8, the Fredholm operators associated to the Dirichlet spaec are +F0 = P0 + D +|D| = +� +p0 +u∗ +u +q0 +� +and F = P0 + +D +√ +1 + D2 = +� +p0 +I +√I+L∂∗ +∂ +I +√I+L +q0 +� +. +They differ by an element in the ideal generated by |D|−2 (cf. subsection 2.1). +Proposition 2.4, Corollary 2.6 and point v) of Theorem 3.8 lead to the following representation +Proposition 3.10. i) For a = a∗ ∈ AE there exist bounded operators α, β and γ such that +i[F, a] = α|D|−1 + |D|−1β + |D|−1/2γ |D|−1/2 +with ||α|| ≤ 2 || [D, a] = 2||R(a)||, ||β|| ≤ 2 ||R(a)||, ||γ|| ≤ || R(a) ||; +ii) µ8k([F, a]) ≤ 5 max(||R(a)|| ||R(a∗||) λ−1/2 +k+1 for all k ∈ N. +Remark 3.11. When the Lipschitz algebra is not dense in A, as in the case of harmonic +forms on the p.c.f. fractals where AE reduces to constant functions, an alternative, natural +choice for the Fredholm module is (H, A, �F) where �F = q⊥ +0 − q0 is the orthogonal symmetry +on H with respect to the subspace of gradients Im(∂) = q0(H). This is what has been done +in [CS2] for post critically finite fractals]. +This alternative choice leads to different estimates for the quantum derivative (cf. [CS2]). +However, up to a sign, they lead to the same K-homology class, as shown in the following +Lemma 3.12. If L has discrete spectrum and AE is dense in A, the Fredholm modules +(L2(A, τ) ⊕ H, A, F) and +� +L2(A, τ) ⊕ H, A, I ⊕ (− �F) +� +are homotopic. +Proof. We forget about p0 which is finite dimensional. (L2(A, τ) ⊕ H, A, F) and (L2(A, τ) ⊕ +H, A, F0) are obviously homotopic (through the family (L2(A, τ) ⊕ H, A, (1 − t)F + t F0), +t ∈ [0, 1], while (L2(A, τ)⊕H, A, F0) and +� +L2(A, τ)⊕H, A, I ⊕(− �F) +� +are homotopic through +the family of Fredholm operators +� +sin θ I +cos θ u∗ +cos θu +(1 + sin θ)q0 − sin θ I +� +θ ∈ [0, π/2]. +□ +3.8. Commutators with elements of the smooth subalgebra. In this subsection, we +start with some selfadjoint element a ∈ A2,∞ +L +. The goal is to prove that the commutator +[ |D|, a ] is bounded, so that Proposition 2.11 applies. +We start with some intermediary +results. +Lemma 3.13. [L, a] +1 +√ +1 + L is a bounded operator. +Proof. Let us compute, for c ∈ A and b ∈ DomL2(L) ∩ B and making use of the rules +established in [CS, square roots] and making use of the symmetry of the A-A-bimodule H : +(c|[L, a]b) = τ(c∗(L(ab) − aL(b))) = τ(c∗(−2Γ(a∗, b) + L(a)b) += −2(∂(a∗)c, ∂b) + (c, L(a)b) += −2(c, R(a∗)∗∂b) + (c, L(a)b) + +16 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +from which we deduce +(3.4) +[L, a] = −2R(a∗)∗∂ + L(a) . +□ +As a consequence, we establish the following: +Lemma 3.14. For γ ∈ (0, 1/2), (I + L)1/2−γ [(I + L)γ, a ] is a bounded operator. +As a consequence, [(I + L)γ, a ] is a compact operator. +Preuve : We start with +� +(1 + L)γ , a +� += Cγ +� +∞ +0 +tγ−1 +� +1 + L +t + 1 + L , a +� +dt += −Cγ +� +∞ +0 +tγ−1 +� +t +t + 1 + L , a +� +dt += Cγ +� +∞ +0 +tγ +1 +t + 1 + L [L, a] +1 +t + 1 + L dt += Cγ +� +∞ +0 +tγ +1 +t + 1 + L [L, a] +1 +√ +1 + L +√ +1 + L +t + 1 + L dt . +Coupling with x, y ∈ L2(A, τ), Lemma 3.13 provides some constant C′ such that : +��� +� +y, +� +(1 + L)γ , a +� +x +���� ≤ C′ +� +∞ +0 +tγ ��� +1 +t + 1 + L y +��� +��� +√ +1 + L +t + 1 + L x +��� dt +≤ C′ +�� +∞ +0 +t2γ� +y , +1 +(t + 1 + L)2 y +� +dt +�1/2 �� +∞ +0 +� +x , +1 + L +(t + 1 + L)2 x +� +dt +�1/2 +. +One checks easily that +� +∞ +0 +t2γ +1 +(t + 1 + L)2dt is proportional to (1 + L)2γ−1, and that +� +∞ +0 +1 + L +(t + 1 + L)2dt is the identity operator. +From which +��� +� +y, +� +(1 + L)γ , a +� +x +���� ≤ C′′||(1 + L)γ−1/2y || ||x|| and the result +□ +As. a corollary, we get +Lemma 3.15. [ +√ +1 + L , a ] is a bounded operator. +Proof. Apply Lemma 3.14 with γ = 1/4 : (I + L)1/4[(I + L)1/4, a] and [(I + L)1/4, a](I + L)1/4 +are bounded operators (for the latter, consider the adjoints). The Leibnitz rule provides +[(1 + L)1/2, a] = (1 + L)1/4 [ (1 + L)1/4, a ] + [(1 + L)1/4, a ] (1 + L)1/4 +which is a bounded operator. +□ +We can now prove the main result of this section : +Proposition 3.16. For a ∈ A2,∞ +L +the operator [ |D|, a] is a bounded. Hence, the conclusions +of Theorem 2.11 hold true for a. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS17 +Proof. Let ∂ = uL1/2 be the polar decomposition of ∂. +As [∂, a] = [u, a| L1/2+u [a, L1/2] and u [a, L1/2] are bounded, [u, a| L1/2 is a bounded operator. +We have now +� +(∂∂∗)1/2, a +� += +� +uL1/2u∗, a +� += [∂, a]u∗ + uL1/2 [u∗, a] += [∂, a]u∗ + u +� +[a∗, u]L1/2�∗ +which is a bounded operator. This ends the proof. +□ +4. Lower bounds on singular values of quantum differentials +In this section we consider conditions providing lower bounds for the singular values µk(i[F, a]). +We also propose general and particular examples where these conditions are satisfied. +4.1. Lower bounds on quantum differentials of Dirichlet spaces. +Proposition 4.1. Suppose that a ∈ A2,∞ +L +satisfies the three conditions : +i) The commutator [ +√ +I + L, a] is a compact operator. +ii) R(a)∗R(a) has a finite dimensional kernel +iii) R(a)∗R(a) is invertible on ker(R(a))⊥. +Then there exists an integer k0 and a constant C(a) such that +µk(i[F, a]) ≥ C(a) (1 + ε(k)) λk+k0(L)−1/2 +where C(a) is the infimum of the spectrum of |R(a)| on ker(R(a))⊥, ε(k) is a sequence tending +to 0 as k → ∞ and k0 = dim(ker(L)) + 1. +Proof. Let us start with an observation : as +� +0 +0 +[∂ +I +√I+L, a] +0 +� += +� +0 +0 +I +0 +� � +p0 +[ +I +√I+L∂∗, a] +[∂ +I +√I+L, a] +q0 +� � +0 +0 +0 +I +� +which provides +µk([F, a]) ≥ µk +� +[∂ +I +√ +I + L2, a] +� +. +We have then +[∂ +I +√ +I + L, a] = [∂, a] +I +√ +I + L − ∂ +I +√ +I + L +�√ +I + L , a +� +I +√ +I + L +i.e., since [∂, a] is equal to R(a), ∂ +I +√ +I + L is bounded and +�√ +I + L , a +� +is compact, we get +[∂ +I +√ +I + L, a] = +� +R(a) + κ +� +I +√ +I + L +with κ compact. Applying Lemma 2.15, the thesis follows. +□ +In the sequel, we show that the conditions of the above result are realistic. In particular, +according to equation 3.3 in Remark 3.7, condition ii) is satisfied whenever E is the Dirichlet +form associated with a proper negative type function ℓ on a discrete group G and a = λ(g), +g ∈ G, ℓ(g) ̸= 0. + +18 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +4.2. Roots of generators. ([CS1]). +Suppose that �E is a (symmetric, regular, completely) Dirichlet form on L2(A, τ) with generator +�L such that A2,∞ +�L +is dense in A, and consequently A�L is dense in A. Fix any β ∈ (0, 1), let E +be the Dirichlet form with generator L = �Lβ (cf. [CS1] Section 10.4). The semigroup e−tL is +called subordinated to the semigroup e−t�L (see [C1]). +Lemma 4.2. We have A2,∞ +�L +⊂ A2,∞ +L +. +Proof. We start with the standard formula for roots of positive operators : +�Lβ = Cβ +� +∞ +0 +tβ−1 +�L +t + �L +dt = Cβ +� +∞ +0 +s−β +�L +I + s�L +ds +(4.1) +with Cβ = sin(βπ)/βπ . +As +I +I + s�L +acts as a completely positive contraction of M, for +a ∈ A2,∞ +�L +we have on one hand +�� +�L +I + s�L +(a) +�� +M| ≤ ∥�L(a)∥M +so that the integral converges in M for s → 0, and on the other hand +�� +�L +I + s�L +(a) +�� +M = 1 +s +��a − +I +I + s�L +(a) +�� +M ≤ 2 +s||a||M +so that the integral converges in M as s → +∞. We have proved A2,∞ +�L +⊂ DM(L). As A2,∞ +�L +is an involutive algebra, for a ∈ A2,∞ +�L +we have also a∗ ∈ DM(L) and a∗a ∈ DM(L), so that +Γ[a] = 1 +2 +� +a∗L(a) + L(a)∗a − L(a∗a) +� +∈ M. Which proves A2,∞ +�L +⊂ AL. +□ +Corollary 4.3. With L = �Lβ as in formula (4.1), there exists an involutive subalgebra A∞ +of A2,∞ +L +dense in A such that, for any a ∈ A∞, the commutator [ +√ +I + L, a] is a compact +operator. +Proof. Take A∞ = A2,∞ +�L +and apply Lemma 3.14 with γ = β/2. +□ +5. Slow conditionally negative type function on discrete groups +In this section we come back to the framework of subsection 3.5. Thus, G is a discrete group +with the Haagerup property, τ is the canonical trace on C∗ +red(G), ℓ is a proper negative type +function on G, L is the operator of multiplication by ℓ in ℓ2(G) = L2(C∗ +red(G), τ) and Eℓ is the +Dirichlet form with generator L. We recall that A∞ is the dense ∗-subalgebra of A = C∗ +red(G) +of the a = � +g∈G a(g)λ(g) with finite support. +5.1. Asymptotic orthogonality for 1-cycles. Let us start with a simple observation: for +fixed g ∈ G we have +ℓ(g−1g′) = ℓ(g) + ℓ(g′) − 2(c(g)|c(g′)) = ℓ(g′) + O(ℓ(g′))1/2 +as g′ → ∞ . +We are going to show that Proposition 4.1 applies to the Dirichlet forms Eℓ provided the +negative type function ℓ satisfies a strengthened form of the above asymptotic estimate. +Definition 5.1. (Slow conditionally negative type functions) A conditionally negative type +function ℓ : G → [0, +∞) is said to be slow if it is proper and if, for any fixed g ∈ G, +(5.1) +ℓ(g−1g′) = ℓ(g′) + o(ℓ(g′))1/2 +as g′ → ∞ . + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS19 +Proposition 5.2. Let ℓ : G → 0, +∞) be slow conditionally negative type function. Then, +Proposition 4.1 applies to the Dirichlet form Eℓ for any a = λ(g) with ℓ(g) ̸= 0 (i.e. all g ∈ G +but a finite number), so that there exists a sequence ε(k) → 0 such that +� +ℓ(g) (1 + ε(k)) λk+1(L)−1/2 ≤ µk +� +i +� +F, λ(g) +�� +≤ 5 +� +ℓ(g) λ[k/8]+1(L)−1/2 . +Proof. One checks easily that for such a = λ(g), [ +√ +I + L, a] = kgλ(g) where kg is the multi- +plication operator by the function g′ → +� +1 + l(g−1g′) − +� +1 + ℓ(g′). Check that +� +1 + l(g−1g′) − +� +1 + ℓ(g′) = +ℓ(g−1g′) − ℓ(g′) +� +1 + l(g−1g′) + +� +1 + ℓ(g′) += +o(ℓ(g′)1/2) +� +1 + l(g−1g′) + +� +1 + ℓ(g′) +tends to 0 as g′ → ∞, so that kg is a compact operator. Condition (i) of Proposition 4.1 is +satisfied. Condition (ii) of Proposition 4.1 is provided by identity (3.3) of Remark 3.7. The +estimates come from Proposition 4.1, Corollary 2.6, Conclusion 5 of Theorem 3.8 and identity +(3.3) of Remark 3.7. +□ +Corollary 5.3. Let �ℓ be a proper conditionally negative type function on G. Then ℓ := �ℓβ +is a slow conditionally negative type function, for an arbitrary β ∈ (0, 1) and Proposition 4.1 +applies to the Dirichlet form Eℓ. +Proof. Fix g and make g′ tend to ∞. As noticed above, one has �ℓ(g−1g′) = �ℓ(g′)+O(�ℓ(g′)1/2) = +�ℓ(g′) +� +1 + O(�ℓ(g′)−1/2� +so that +ℓ(g−1g′)) = �ℓ(g−1g′)β = �ℓ(g′)β(1 + O(�ℓ(g′)−1/2) = ℓ(g′) + o(ℓ(g′)) . +□ +Corollary 5.4. Suppose that G = Fp is the free group with p generators and ℓ is the length +function, which is conditionally negative (cf. [Haa]). Then Proposition 4.1 applies to the +Dirichlet form Eℓ for a = λ(g) with g ̸= e so that there exist constants C1, C2 > 0 and an +integer k0 such that +C1(Log(k))−1/2 ≤ µk(i[F, a]) ≤ C2(Log(k))−1/2 , k ≥ k0 +with C1 (resp. C2) arbitrarily close to +� +ℓ(g) (resp. 5 +� +ℓ(g)) if K0 is chosen large enough. +Proof. The first assumption is straightforward: since ℓ is a length function, we have |ℓ(s−1t)− +ℓ(t)| ≤ ℓ(s) so that ℓ is a proper, slow, conditionally negative type function. For the second +assumption, check that the ball Bm of radius m in G, Bm = {g ∈ G | ℓ(g) ≤ m}, has +a cardinality |Bm| = 1 + +p +p − 1(2p − 1)m. +As λk = m for |Bk| ≤ m < ∥Bk+1|, we get +λk ∼ +k +Log(2p − 1), k → ∞. Corollary 2.6 and Proposition 5.2 provide the result. +□ +5.2. Weight functions as slow conditionally negative type functions. Here we prove +that all weight functions on discrete groups are slow negative type functions. +Let ℓ be a conditionally negative type function on the group G (symmetric and strictly +positive away from the unit) and let (hℓ, πℓ, c) be the associated 1-cocycle c : G → hℓ with +(5.2) +c(st) = c(s) + πℓ(s)c(t) , +||c(t)||2 = ℓ(t) , +(c(s)|c(t))H = 1 +2 +� +ℓ(s) + ℓ(t) − ℓ(s−1t) +� +. + +20 +FABIO E.G. CIPRIANI, JEAN-LUC SAUVAGEOT +One checks easily that the vectors c(s), c(t) in hℓ are orthogonal if and only if e lies between +s and t, i.e. ℓ(s−1t) = ℓ(s) + ℓ(t). The function +√ +ℓ is conditionally negative too and provides +a left-invariant metric on G by +d√ +ℓ(s, t) := +� +ℓ(s−1t) +s, t ∈ G . +The cocycle is an isometric embedding of the metric space (G, d√ +ℓ) into the Hilbert space hℓ +∥c(t) − c(s)∥hℓ = ∥c(ss−1t) − c(s)∥hℓ = ∥c(s) + πℓ(s)(c(s−1t)) − c(s)∥hℓ += ∥c(s−1t))∥hℓ = +� +ℓ(s−1t) = d√ +ℓ(s, t) +s, t ∈ G . +The characteristic property of a slow negative type function, expressed as +ℓ(s) + ℓ(t) − ℓ(s−1t) = o( +� +ℓ(t)) +t → ∞, +for each fixed s ∈ G, can be restated as +0 = lim +t→∞ +ℓ(s) + ℓ(t) − ℓ(s−1t) +2 +� +ℓ(s) +� +ℓ(t) += lim +t→∞ +(c(s)|c(t))hℓ +∥c(s)∥Hℓ · ∥c(t)∥hℓ +, +s ∈ G. +The property thus refers to the asymptotic orthogonality of t ∈ G with respect to any fixed +s ∈ G, when these are embedded in the Hilbert space hℓ. +We recall that a weight on a discrete group G ([deH]) is a negative type function such that +ℓ : G → [0, +∞) +ℓ(st) ≤ ℓ(s) + ℓ(t). +Lemma 5.5. Any proper weight function on a discrete group G is a slow, conditionally +negative negative type function. +Proof. Since a weight satisfies |ℓ(s) − ℓ(t)| ≤ ℓ(s−1t), we have 0 ≤ ℓ(s) + ℓ(t) − ℓ(s−1t) ≤ +2(ℓ(s) ∧ ℓ(t)) and +ℓ(s) + ℓ(t) − ℓ(s−1t) +2 +� +ℓ(s) +� +ℓ(t) +≤ +ℓ(s) ∧ ℓ(t) +� +ℓ(s) +� +ℓ(t) +. +Since ℓ is proper we have +lim +t→∞ +ℓ(s) + ℓ(t) − ℓ(s−1t) +2 +� +ℓ(s) +� +ℓ(t) +≤ lim +t→∞ +ℓ(s) ∧ ℓ(t) +� +ℓ(s) +� +ℓ(t) += lim +t→∞ +� +ℓ(s) +ℓ(t) = 0 . +□ +Remark 5.6. A weight function gives rise to a left-invariant metric on G: dℓ(s, t) := ℓ(s−1t). +If k := inf{ℓ(t) : t ∈ G, t ̸= e}, the cocycle is a Lipschitz embedding of the metric space +(G, dℓ) into the real Hilbert space hℓ +∥c(t) − c(s)∥H = +� +ℓ(s−1t) ≤ +� +k−1ℓ2(s−1t) = +1 +√ +k +ℓ(s−1t) = +1 +√ +k +dℓ(s, t) +s, t ∈ G . +Examples include length functions of free groups Fn and the Heisenberg group. +REFERENCES +[AHK] S. Albeverio, R. Hoegh-Krohn, Dirichlet forms and Markovian semigroups on C∗– +algebras, Comm. Math. Phys. 56 (1977), 173-187. +[C1] F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von +Neumann algebras, J. Funct. Anal. 147 (1997), 259-300. +[C2] F. Cipriani, “Dirichlet forms on Noncommutative spaces”, Springer ed. L.N.M. 1954, +2007. + +QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS21 +[CFK] F. Cipriani, U. Franz, A. Kula, Symmetries of Lévy processes on compact quantum +groups, their Markov semigroups and potential theory, J. Funct. Anal. 266 (2014), +no. 5, 2789–2844. +[CS1] F. Cipriani, J.-L. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. +Anal. 201 (2003), no. 1, 78–120. +[CS2] F. Cipriani, J.-L. Sauvageot, Fredholm modules on P.C.F. self-similar fractals and +their conformal geometry, Comm. Math. Phys. 286 (2009), no. 2, 541–558. +[CS3] F. Cipriani, J.-L. Sauvageot, Measurability, spectral densities and hypertraces in Non- +commutative Geometry, to appear in Journal of Noncommutative Geometry. +[Co] A. Connes, “Noncommutative Geometry”, Academic Press, New York, 1994. +[DL] E.B. Davies, J.M. Lindsay, Non–commutative symmetric Markov semigroups, Math. +Z. 210 (1992), 379-411. +[deH] P. de la Harpe, “Topics in Geometric Group Theory”, +Chicago Lectures in Mathematics, The University of Chicago Press, 2000. +[GK] I.C. Gohberg, M.G. Krein, “Introduction to the theory of linear nonselfadjoint opera- +tors”, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, R.I., 1969;. +[Haa] U. Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approxi- +mation property, Invent. Math. 50 (1978), no. 3, 279-293. +[Hil] M. Hilsum, Signature operator on Lipschitz manifolds and unbounded Kasparov bi- +modules, Lecture Notes in Math., 1132 Operator algebras and their connections with +topology and ergodic theory (Busteni, 1983)(1985), 254-288. +[S] J.-L. Sauvageot, Quantum Dirichlet forms, differential calculus and semigroups, Quan- +tum Probability and Applications V, Lecture Notes in Math. 1442 (1990), 334-346 +[SWW] E. Schrhoe, M. Walze, J.-M. Warzecha, Construction de triplets spectraux a partir de +modules de Fredholm, C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), 1195–1199. +[V] D. Voiculescu, On the existence of quasicentral approximate units relative to normed +ideals. I., J. Funct. Anal. 91 (1990), no. 1, 1-36. +(F.E.G.C.) Politecnico di Milano, Dipartimento di Matematica, piazza Leonardo da Vinci 32, +20133 Milano, Italy. +Email address: fabio.cipriani@polimi.it +(JLS) Institut de Mathématiques de Jussieu – Paris Rive Gauche, CNRS – Université Paris +Cité, F-75205 Paris Cedex 13, France +Email address: jean-luc.sauvageot@imj-prg.fr + diff --git a/R9AyT4oBgHgl3EQfVPdi/content/tmp_files/load_file.txt b/R9AyT4oBgHgl3EQfVPdi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b5529a077e0e81aa8a02f76d547ae9b8db0260c --- /dev/null +++ b/R9AyT4oBgHgl3EQfVPdi/content/tmp_files/load_file.txt @@ -0,0 +1,781 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf,len=780 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='00140v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='OA] 31 Dec 2022 1 QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We study natural conditions on essentially discrete spectral triples (A, h, D) by which the quantum differential da of a ∈ A belongs to the ideal generated by the unit length ds = D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We also study upper and lower bounds on the singular values of the da’s and apply the general framework to natural spectral triples of Dirichlet spaces and, in particular, those on dual of discrete groups arising from negative definite functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Introduction In Noncommutative Geometry [Co] a spectral triple (A, h, D) is made by a ∗-algebra of bounded operators A ⊆ B(h) on a Hilbert space h on which a densely defined self-adjoint (Dirac) operator D also acts in such a way that the commutators [D, a] ∈ B(h) are bounded for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The unit length or line element of (A, h, D) ([Co] Chapter 7), given by the compact operator ds = D−1, is meant to encode the NC metric aspects of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The volume element, given by the spectral density operator dv = ρ(D) [CS3], where ρ(x) := N|D|(x)−1 is the reciprocal of the counting function of |D|, represents the NC measure theoretic aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' It can be written as dv = σ(ds) where σ(x) := N|D|(x−1)−1 is the spectral density function of the compact operator ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Other fundamental infinitesimals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' compact operators) are the quantum differentials da := i[F, a] of elements a ∈ A, where (A, h, F) is a Fredholm module associated to the spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the classical case of the Riemannian geometry of a compact smooth manifold, where A is the algebra of smooth functions, D is the Dirac operator and F is the Hilbert transform, connections among the da’s, ds and their singular values, can be established thanks to the tools of the pseudo-differential calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Even in this commutative setting, however, a much finer analysis is needed when one only requires A to be an algebra of Lipschitz or Sobolev functions, as in the case of the one dimensional fractals and quasi- conformal manifolds (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' works of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Connes-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Sullivan in [Co] and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Hilsum in [Hil]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As a rule, the higher smoothness of an element a ∈ A has to be read by the stronger compactness of its quantum differential da which, in ultimate analysis, will lie in the ideal generated by the unit length ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In this work we estimates the singular values of the da′s in terms of those of ds without imposing any spectral growth condition on the Dirac operator D and only requiring minimal and natural smoothness: A is a subalgebra either in the domain AD of the generator of the automorphisms group α(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' t) := eitDae−itD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' or it is lying in the intersection AD ∩ A|D| with the domain A|D| of the generator of the automorphisms group β(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' t) := eit|D|ae−it|D| (in other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' the commutators with D or |D| are bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In Quantum Mechanics α represents the Heisenberg time evolution of observables in a system governed by the Dirac Hamiltonian 1This work has been supported by Laboratoire Ypatia des Sciences Mathématiques C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' France - Laboratorio Ypatia delle Scienze Matematiche I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Italy (LYSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Date: December 30, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 58B34, 31C25, 46L57, 47A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 1 2 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT D while, in Riemannian Geometry, β represents the geodesic flow, by the Egorov Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In order to deal with more examples, among which the forthcoming ones, we allow the Dirac operator to have an infinite dimensional kernel, while it is assumed to have discrete spectrum away from its kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' to have compact inverse D−1 on the orthogonal ker(D)⊥ of its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' To the range of application of NCG, we add, with this work, Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' These are C∗ or von Neumann algebras, commutative or not, with a privileged quadratic form E satisfy- ing a characteristic contraction property and named Dirichlet form, representing an energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This functional generalizes the Dirichlet integral of a Riemannian manifold and allows to develop a kernel-free NC Potential Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In a Dirichlet space, the Dirac operator is a 2 × 2 matrix D = � 0 ∂∗ ∂ 0 � defined by the derivation ∂, which represents the differential square root of the Dirichlet energy form (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [CS1]) in the sense that E[a] = ∥∂a∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Natural examples come from ground state representations of Hamiltonian in Quantum Me- chanics, completely positive, Markovian semigroups converging to KMS equilibria in Quan- tum Statistical Mechanics and generators of quantum Levy processes in Quantum Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the final part of the work we study, in particular, quantum differentials of Dirichlet spaces associated to negative definite functions on countable, discrete groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' A somewhat detailed account of the content of the work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In Section 2, we introduce essentially discrete spectral triples (A, h, D), a slight enlargement of the classical notion which will be needed for applications to Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the section, we consider a canonical Fredholm operator F0 and a technical variant F of it, associated to any of these triples, showing that they furnish equivalent Fredholm modules on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The analysis is based on old and new representations of the quantum differentials da in terms of the gradient i[D, a] and the line element ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In particular, it is shown that the da’s belong to the ideal generated by � |ds| and that the singular values µ4k(da) are controlled by the singular values µk(ds) with ratios proportional to the Lipschitz seminorms ∥[D, a]∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Assuming that also the commutators [|D|, a] are bounded, we prove that the da’s belong to the ideal generated by ds and that the µ2k(da)’s are dominated by the µk(ds)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Their ratios are shown to be asymptotically close to five times the sum of the seminorms ∥[D, a]∥, ∥[|D|, a]∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In case of discrete spectral triples the estimate is improved to control µk+d(da) where d := dim ker(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In Section 3, we introduce the canonical, essentially discrete spectral triple and Fredholm module of a Dirichlet space (E, F) on a C∗-algebra with l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' faithful trace (A, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The Dirac operator is here the anti-diagonal matrix of the derivation ∂ and divergence ∂∗ canonically associated to E, and A is the Lipschitz algebra AE made by elements of the Dirichlet algebra B := F ∩ A which have bounded carré du champ (alias energy density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Elements of the smooth sub-algebra A2,∞ E ⊆ AE, which belong to the domain of the generator L of E on the von Neumann algebra M := L∞(A, τ), are shown to have bounded commutators with |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' These results are consequences of the compactness on L2(A, τ) of the commutators with the roots (I + L)γ when γ ∈ (0, 1/2) and their boundedness when γ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In Section 4 we prove lower bounds on the singular values of a quantum differential da when the commutator [ √ I + L, a] is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In this case, the singular values of the da’s are controlled both above and below by those of the line element ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The case of Dirichlet forms whose generator are roots Lβ by β ∈ (0, 1) of L is especially considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS3 In the final Section 5, we analyze Dirichlet forms constructed by negative type functions on discrete groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We introduce the class of slow negative type functions to which the entire above analysis apply, showing also that it contains the length function of free groups and any root of any proper negative type function on any discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Essentially discrete spectral triples and their Fredholm modules To cover applications to Dirichlet spaces, we consider spectral triples of the following type: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' (A, h, D) is said essentially discrete if σ(D) \\ {0} is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Thus, if 0 ∈ σ(D), this value is allowed to be an eigenvalue with infinite degeneracy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=', denoting by P0 the orthogonal projection onto ker (D), we are assuming that (I − P0)D has discrete spectrum as an operator on (I − P0)(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We shall adopt the convention by which D−1 and |D|−1 denote the operators which identically vanish on ker (D) = P0h and are the usual functional calculi on (I − P0)(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' With this convention, both these operators are compact and we have the identities DD−1 = D−1D = |D| |D|−1 = |D|−1|D| = I − P0, D|D|−1 = sign(D) where the last one is the sign operator corresponding to the sign function on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The nonzero part of the spectrum of |D| will be enumerated as 0 < λ1(|D|) ≤ λ2(|D|) · · · ≤ λn(|D|) ≤ λn+1(|D|) ≤ · · · where each eigenvalue λn(|D|) is repeated according to its multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Fredholm operators and quantum differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the commutative situation where D is the Dirac operator on the Clifford algebra of a closed, compact, Riemannian manifold M, the sign operator Fcl := sign(D) is a 0-order ΨDO, since D is differential operator and |D| is a ΨDO both of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' By the pseudo-differential calculus, the com- mutators [Fcl, a] with smooth functions a ∈ C∞(M) are ΨDO of order −1 so that [Fcl, a]|D| and |D|[Fcl, a] are bounded as 0-order ΨDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In other words, the commutators [Fcl, a] with smooth functions belong to the symmetric principal ideal (of compact operators) generated by the line element ds = |D|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This property cannot be derived if the functions a are just Lipschitz because in these cases pseudo-differential calculus does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The purpose of this section is to get similar estimates in the general noncommutative geometrical workframe under rather mild assumptions on a ∈ B(h) of Lipschitz nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' To consider the general situation, we associate to the Dirac operator, the following self-adjoint, bounded operator F := P0 + D √ 1 + D2 which is in fact a Fredholm one, as it follows from F 2 − I = P0 + (I − P0) D2 I + D2 − Ih = −(I − P0) I I + D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' An alternative self-adjoint, Fredholm operator will be also considered: F0 := P0 + D |D|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' It is an orthogonal symmetry which is equal to F up to a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' More precisely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1) F 2 = I, F − F0 = (I − P0)T |D|−2 4 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT where T := − D |D| D2 √ I + D2(|D| + √ I + D2) is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The operators i[F0, a], i[F, a], both denoted by da, are the quantum differentials of a ∈ A ([Co] Chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us start with an observation: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The following identity holds true [P0, a] = P0 α0 |D|−1 + |D|−1β0 P0 a ∈ A for suitable operators α0, β0 ∈ B(h) with ∥α0∥ = ∥β0∥ = ∥[D, a]∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The stated representation follows from the identities P0a − P0aP0 = P0a(I − P0) = P0aDD−1 = P0[a, D]D−1 = P0[a, D]|D|D−1|D|−1 P0aP0 − aP0 = (I − P0)aP0 = D−1DaP0 = D−1[D, a]P0 = |D|−1|D|D−1[D, a]P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For a = a∗ ∈ A we have the bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2) − || [D, a] || (I + D2)−1/2 ≤ i � D √ 1 + D2, a � ≤ || [D, a] || (I + D2)−1/2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) − || [D, a] || |D|−1 ≤ i (I − P0) � D √ 1 + D2, a � (I − P0) ≤ || [D, a] || |D|−1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4) − ∥[D, a]∥ · |D|−1 ≤ i (I − P0) � D |D|, a � (I − P0) ≤ ∥[D, a]∥ · |D|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice that the double inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) is a straightforward consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2), since (I − P0) 1 √ 1 + D2(I − P0) ≤ |D|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In order to prove the double inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' we use,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' as usual (see [SWW] Proposition 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' the identity (in the strongly convergent sense) D √ I + D2 = 1 π � +∞ 0 t−1/2 D tI + I + D2dt QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' DIRICHLET SPACES AND DISCRETE GROUPS5 from which we deduce i � D √ 1 + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � = 1 π � +∞ 0 t−1/2 i � D tI + I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � dt = 1 π � +∞ 0 t−1/2 � I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] + I � I tI + I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � D � dt = 1 π � +∞ 0 t−1/2 � I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] − I tI + I + D2i[D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] D tI + I + D2 � dt = 1 π � +∞ 0 t−1/2� I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] − D tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] D tI + I + D2 − I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] D2 tI + I + D2 � dt = 1 π � +∞ 0 t−1/2� I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] � I − D2 tI + I + D2 � − D tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] D tI + I + D2 � dt = 1 π � +∞ 0 t−1/2� tI + I tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] I tI + I + D2 − D tI + I + D2i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] D tI + I + D2 � dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Noticing now that i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] is self-adjoint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' so that −|| [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || I ≤ i[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] ≤ || [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' the above identity thus provides i � D √ 1 + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � ≤ || [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || 1 π � +∞ 0 t−1/2� tI + I (tI + I + D2)2 + D2 (tI + I + D2)2 � dt = || [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || 1 π � +∞ 0 t−1/2 I tI + I + D2 dt = || [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || (I + D2)−1/2 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' i � D √ 1 + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � ≥ −|| [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || 1 π � +∞ 0 t−1/2� tI + I (tI + I + D2)2 + D2 (tI + I + D2)2 � dt = −|| [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || 1 π � +∞ 0 t−1/2 I tI + I + D2 dt = −|| [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] || (I + D2)−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The double inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4) can be proved the same way, with obvious adaptations, or derived from Proposition 1 in [SWW] where the same inequality is stated in case D is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In fact, on one hand i (I−P0) � D |D|, a � (I−P0) = i � (I−P0) D |D| (I−P0) , (I−P0)a(I−P0) � = i � D |D|, (I−P0)a(I−P0) � and, since ∥ � D, (I −P0)a(I −P0) � ∥ = ∥[D, a]∥, (D, (I −P0)h, (I −P0)A(I −P0)) is a spectral triple where D is invertible on (I − P0)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 6 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT Using the bounds above, we show that the quantum differentials of elements of A belong to the symmetric principal ideal in B(h) generated by � |ds| = |D|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For any fixed a = a∗ ∈ A i) there exist bounded operators α, β, γ ∈ B(h) such that i[F, a] = α|D|−1 + |D|−1β + |D|−1/2γ |D|−1/2, with ||α|| ≤ 2 || [D, a] ||, ||β|| ≤ 2 || [D, a] ||, ||γ|| ≤ || [D, a] ||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) a similar representation holds true with F0 in place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' i) For [P0, a], see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Since DP0 = 0 we have P0[(1 + D2)−1/2D, a] = −P0aD(1 + D2)−1/2(I − P0) = P0[D, a](1 + D2)−1/2(I − P0) = P0[D, a] |D| √ 1 + D2|D|−1 = P0[D, a] |D| √ 1 + D2|D|−1(1 − P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Similarly, [(1 + D2)−1/2D, a]P0 = (1 − P0)|D|−1 |D| √ 1 + D2 [D, a]P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As for (I −P0)[(1+D2)−1/2D, a](I −P0) = (I −P0)[(1+D2)−1/2D, (I −P0)a(I −P0)](I −P0), one invokes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3, which is equivalent to the assertion −|| [D, a] || |D|−1 ≤ i � D √ I + D2, (I − P0)a(I − P0) � ≤ −|| [D, a] || |D|−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3 and the same proof as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ As first consequence of the above representation, we have Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' If (A, h, D) is an essentially discrete spectral triple, then (A, h, F0) and (A, h, F) are, essentially unitary equivalent, Fredholm modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Follows from the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ A second consequence concerns a bound on the singular values of the quantum differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' If (A, h, D) is an essentially discrete spectral triple, the singular values of quantum differentials are controlled by the Lipschitz seminorm and the singular values of D−1 µ4k(i[F0, a]) ≤ 5 ∥ [D, a] ∥ µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 a = a∗ ∈ A, k ≥ 0, µ4k(i[F, a]) ≤ 5 ∥ [D, a] ∥ µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 a = a∗ ∈ A, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In terms of line element and quantum differentials, we proved the bounds µ4k(da) ≤ 5 ∥ [D, a] ∥ µk(ds) a = a∗ ∈ A, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4 and the rules of singular values [Co] Chapter 4 Appendix C, we have µ4k(i[F, a]) ≤ µk(α|D|−1) + µk(|D|−1β) + µ2k(|D|−1/2γ |D|−1/2) ≤ ∥α∥µk(|D|−1) + ∥β∥µk(|D|−1) + ∥γ∥µk(|D|−1/2)2 ≤ 5 ∥ [D, a] , µk(|D|−1) = 5 ∥ [D, a] ∥ λk+1(|D|)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' and the same for µ4k(i[F0, a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Improving the estimates for the singular values of quantum differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Here we consider a natural condition by which the quantum differentials da of elements a ∈ A belong to the principal ideal in K(h) generated by the line element ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The quantum differ- entiation operator d can then be considered as a derivation from A to the ideal ID ⊆ K(h) generated by ds, seen as a A-bimodule over the norm closure A := A ⊆ B(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then one has [P0, a](I − P0) = P0[a, D]F0 |D|−1 � D |D|, a � (I − P0) = −D|D|−1[ |D|, a] |D|−1 + [D, a] |D|−1 � D √ I + D2, a � (I − P0) = − D √ I + D2 �√ I + D2, a � |D| √ I + D2|D|−1 + [D, a] |D| √ I + D2|D|−1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5) and [P0, a]P0 = −|D|−1F0[D, a]P0 � D |D|, a � P0 = |D|−1[D, a]P0 � D √ I + D2, a � P0 = |D|−1 D √ I + D2[D, a]P0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us compute successively : [P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a](I − P0) = P0a(I − P0) = P0aDD−1 = P0[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' D]D−1 = P0[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' D]F0|D|−1 � D |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) = D[ |D|−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a](I − P0) + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D|−1 = D(I − P0)[ |D|−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a](I − P0) + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D|−1 = −D|D|−1[ |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D|−1 + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D|−1 � D √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) = D � I √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] I − P0 √ I + D2 = D(I − P0) � |D|−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D|−1 = −D I √ I + D2 � √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � I − P0 √ I + D2 + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] I − P0 √ I + D2 = −D I √ I + D2 � √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � ID| √ I + D2 |D|−1 + [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] |D| √ I + D2 |D|−1 [P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a]P0 = −(I − P0)aP0 = −D−1[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a]P0 = −|D|−1F0[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a]P0 � D |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � P0 = D |D|aP0 = |D|−1[D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a]P0 � D √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � P0 = D √ I + D2aP0 = I − P0 √ I + D2 � D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � P0 = |D|−1 |D| √ I + D2 [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a] P0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 8 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then there exist bounded operators α0 and β0 such that i[F0, a] = α0 |D|−1 + |D|−1 β0 with ∥α0∥ ≤ ∥ [D, a] ∥, ∥β0∥ ≤ ∥ [D, a] ∥+∥ [ |D|, a ] ∥ and β0P0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The quantum differential da = i[F0, a] thus belongs to the symmetric ideal ID ⊆ B(h) generated by the line element ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This is a straightforward consequence of the previous Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ The similar result for the commutator [F, a] needs a preliminary lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [|D|, a] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then i) the following bound holds true ∥[ √ 1 + D2, a] ∥ ≤ ∥[|D|, a] ∥ + 2∥a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) one has ∥(I − P0) �√ 1 + D2, a � (I − P0)∥ ≤ C1(λ1)∥ � |D|, a � ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' where C1(λ1) = � 1 + λ−2 1 and λ1 := λ1(|D|) > 0 is the first nonzero eigenvalue of |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The estimate in i) follows from the identity √ 1 + D2 = |D| + I √ 1 + D2 + |D| and the bound ∥( √ 1 + D2 + |D|)−1∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The same identity also implies (I − P0 � [ √ 1 + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) =(I − P0) � |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) − I − P0 √ 1 + D2 + |D| � √ 1 + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � I − P0 √ 1 + D2 + |D| − I − P0 √ 1 + D2 + |D| � |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � I − P0 √ 1 + D2 + |D| and the inequality �� (I − P0)[ √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) �� ≤ �� [ |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a ��� + ( � 1 + λ2 1 + λ1)−2�� (I − P0)[ |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) �� +( � 1 + λ2 1 + λ1)−2�� (I − P0)[ √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) �� which in turn provides � 1 − ( � 1 + λ2 1 + λ1)−2��� (I − P0)[ √ I + D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � (I − P0) �� ≤ � 1 + ( � 1 + λ2 1 + λ1)−2� ∥[ |D|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a � ∥ and the result with C1(λ1) = 1 + ( � 1 + λ2 1 + λ1)−2 1 − ( � 1 + λ2 1 + λ1)−2 = 1 + ( � 1 + λ2 1 − λ1)2 1 − ( � 1 + λ2 1 − λ1)2 = 1 + λ2 1 − λ1 � 1 + λ2 1 λ1 � 1 + λ2 1 − λ2 1 = 1 λ1 � 1 + λ2 1 − λ2 1 − 1 = � 1 + λ2 1 + λ1 λ1 − 1 = � 1 + λ−2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' One can slightly improve estimate i) above by replacing ||a|| by the norm of a in the quotient space A/A ∩ {D}′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS9 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then there exist bounded operators α1, β1 ∈ B(h) such that i[F, a] = α1|D|−1 + |D|−1β1 with ∥α1∥ ≤ 2∥ [D, a] ∥, ∥β1∥ ≤ 3∥ [D, a] ∥+C1(λ1)|| [ |D|, a] ∥ (or, at choice, ∥β1∥ ≤ 3∥ [D, a ]∥+ ∥ [ |D|, a] ∥ + 2||a||) and β1 = β1 P0 and the quantum differential da = i[F, a] belongs to the symmetric ideal ID ⊆ B(h) generated by the line element ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' A straightforward consequence of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded, then singular values of the quantum differentials are controlled by µ2k(i[F0, a]) ≤ (||α0|| + ||β0||) µk(|D|−1) = (||α0|| + ||β0||) ∥ λk+1(|D|)−1 µ2k(i[F, a]) ≤ (||α1|| + ||β1||) µk(|D|−1) = (||α1|| + ||β1||) ∥ λk+1(|D|)−1 with α0, β0, α2 and β1 provided by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In terms of line element and quantum differentials, we have µ2k(da) ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='µk(ds) a = a∗ ∈ A, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Apply Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11 along the lines of the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ From this proposition, in case of discrete spectrum, we deduce the following : Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A, a = a∗ such that commutator [ |D|, a] is bounded and suppose that the kernel of D is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then one has the estimates µk+d(i[F0, a]) ≤ ||α0|| µk(|D|−1) = ||α0|| λk+1(|D|)−1 µk+d(i[F, a]) ≤ ||α1|| µk(|D|−1) = ||α1|| λk+1(|D|)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' with α0 and α1 are provided by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11 respectively and d = dim(ker(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In particular, ||α0|| ≤ || [D, a] || and ||α1|| ≤ 2|| [D, a] ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In terms of line element and quantum differentials, we have µk+d(da) ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='µk(ds) a = a∗ ∈ A, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The same estimates from an asymptotic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Here we prove under the double Lipschitz assumptions above, estimates which are asymptotically a bit more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then there exist bounded operators α3, β3, γ3 ∈ B(h) such that i[F, a] = α3|D|−1 + |D|−1β3 + |D|−1γ3|D|−1 with ∥α3∥ ≤ 2∥ [D, a] ∥, ∥β3∥ ≤ 3∥ [D, a] ∥ + || [ |D|, a] ∥ and γ3 bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7, we have [P0, a] = σ0|D|−1 + |D|−1τ0 with ||σ0|| ≤ || [D, a] || ||τ0|| ≤ || [D, a] || � D √ I + D2, a � P0 = |D|−1τ1 with ||τ1|| ≤ || [D, a] || � D √ I + D2, a �� I − P0 � = σ1|D|−1 − D √ I + D2[ √ I + D2, a] D √ I + D2 with ||σ1|| ≤ || [ |D|, a] ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 10 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT We compute [ √ I + D2, a] = [ |D|, a] + � I √ I + D2 + |D| , a � = [ |D|, a] − I √ I + D2 + |D| � √ I + D2 + |D|, a � I √ I + D2 + |D| and notice that � √ I + D2+|D|, a � is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Summing up, we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ We need a Lemma which slightly ameliorates a result due to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Fan ([GK] Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' II par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For sake of completeness we provide a detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let T and σ be two compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then there exist an integer d1 and two sequences (εk)k≥0 and (ε′ k)k≥0 such that limk→∞ εk = limk→∞ ε′ k = 0 and (1 − ε′ k)µk+d1(T) ≤ µk � T(I + σ) � ≤ (1 + εk)µk(T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us recall ([Connes chapter 4 section 2]) that µk(T) = inf ||P ⊥T|| = inf ||T Q⊥|| where P or Q runs in the set of orthogonal projections with rank less than k, and that the infimum is indeed a minimum, reached when P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Q) is the orthogonal projection corresponding to the k first larger eigenvalues of |T ∗| (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' |T|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let Pk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Qk) be the orthogonal projection corresponding the k first eigenvalues of |T ∗| (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' |T|) : we have µk(T) = ||P ⊥ k T|| and PkT = TQk, hence P ⊥ k T = TQ⊥ k = P ⊥ k TQ⊥ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice that, as k → ∞, the Qk tend increasingly toward I − q0, so that the Q⊥ k tend to q0, where q0 is the orthogonal projection on the kernel of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice that, as σ is compact, limk→∞ ||Q⊥ k (I − q0)σ|| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Compute now µk � T(I + σ) � ≤ ||P ⊥ k T(I + σ)|| = ||P ⊥ k TQ⊥ k (I − q0)(I + σ)|| ≤ ||P ⊥ k T|| � 1 + ||Q⊥ k (I − q0)σ||) which provides the right inequality, with εk = ||Q⊥ k (I − q0)σ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice now that there exists a compact operator τ such that (I + σ)(I + τ) = I − p1, where p1 is the orthogonal projection on ker(I + σ∗) = Im(I + σ)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This is a finite rank projection, with rank d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Applying the inequality just proved above, we can write µk+d1(T) = µk+d1 � T(I − p1) + Tp1 � ≤ µk � T(I − p1) � + µd1(Tp1) = µk � T(I − p1) � = µk � T(I + σ)(I + τ) � ≤ µk � T(I + σ) � (1 + �εk) with limk→∞ �εk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let a ∈ A be such that the commutator [ |D|, a ] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then one has µ2k � i[F, a] � ≤ � 5|| [D, a] || + || |D|, a] || � (1 + εk) λk+1(|D|)−1 with lim k→∞ εk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Spectral triples of Dirichlet spaces In this section we construct the spectral triple of a Dirichlet space on a C∗-algebra with trace and we apply the previous results to study the associated Fredholm module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Dirichlet forms and their tangent bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For the definition and properties of Dirichlet forms on trace C∗-algebras we refer to [AHK], [C1], [C2], [CS1], [DL], [S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We list below their main properties we need and fix notations for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the following, (A, τ) is a C∗-algebra equipped with a faithful, densely defined, lower semi- continuous trace and M := πτ(A)′′ ⊆ B(L2(A, τ)) is the corresponding von Neumann algebra acting on the Hilbert space of the GNS representation πτ : A → B(L2(A, τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We consider a completely Dirichlet form (E, F) on L2(A, τ) and its densely defined, positive, self-adjoint generator (L, D(L)) in such a way that E is the closure of the quadratic form D(L) ∋ ξ → (ξ|Lξ) and one has F = D(L1/2) and E[ξ] = ∥L1/2ξ∥2 for ξ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Since now on, we shall assume that (L, D(L)) has discrete spectrum away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Among the characteristic properties of a Dirichlet form, we recall that i) the semigroup {e−tL : t > 0} maps L2(A, τ) ∩ M into itself and extends to a σ-weakly continuous, completely positive contraction semigroup of M, still denoted by the same symbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) the resolvent {(I + tL)−1 : t > 0} maps L2(A, τ) ∩ M into itself and extends to a σ-weakly continuous, completely positive contraction resolvent of M, still denoted by the same symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The generator of the semigroup on M, denoted by (L, DM(L)), has a domain DM(L) := {x ∈ M : L(x) := lim t↓0 (x − e−tLx)/t exists σ − weakly in M} which, for any t > 0, coincides with (I + tL)−1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The Dirichlet algebra B := F ∩ A is an involutive subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We assume that (E, F) is regular in the sense that B is dense both in A and L2(A, τ), in their respective topologies, and that it is a form core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Tangent bimodule of a Dirichlet space and carré du champ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us recall the main results of [CS1]: to any regular Dirichlet form (E, F) is associated a symmetric A- bimodule (H, J ) together with a symmetric derivation ∂ : B → H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a linear map satisfying ∂(a∗) = J (∂a) a, b ∈ B, and the Leibnitz rule (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1) ∂(ab) = a∂b + (∂a)b a, b ∈ B, which is closable as a densely defined operator from L2(A, τ) into H, and such that E[a] = ∥∂a∥2 H a ∈ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The carré du champ or energy density of a ∈ B is the following positive linear form Γ[a] ∈ A∗ + ⟨Γ[a], b⟩ = (∂a|(∂a)b)H b ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' A useful approximation of the carré du champ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [CS1]) is the following one: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' i) For any ε > 0, Lε := L 1 + εL generates a bounded Dirichlet form on L2(A, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) Setting Γε[a] := 1 2 � a∗Lε(a) + Le(a)∗a − Lε(a∗a) � ∈ M ∩ L1(A, τ) for a ∈ B, one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2) ⟨Γ[a], b⟩ = lim ε↓0 τ � Γε[a] b � b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 12 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' i) A regular Dirichlet form provides the C∗-algebra of a potential theoretic structure which generalizes the classical Dirichlet integral on a Riemannian manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) the derivation (∂, B) is closable and the domain of its closure coincides with the form domain F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' When no confusion can arise, we shall use the same notation ∂ for the closure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' iii) there is not, in general, a formula for the adjoint divergence operator ∂∗ from H to L2(A, τ), except in case where there exists a subalgebra B0 of A contained in the domain of L, for which ∂∗(∂(a)b) = 1 2 � L(ab) + L(a)b − aL(b) � , a, b ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In the classical case of the Dirichlet integral, the derivation coincides with the gradient oper- ator and its adjoint with the divergence operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' An example of computation of derivation and divergence when any such subalgebras B0 trivialize to the multiples of 1A, is given on fractals in [CGIS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The Lipschitz algebra of a Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Here we isolate a subalgebra of the Dirichlet algebra B which play the role of algebra of Lipschitz functions on a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For a ∈ B, the following conditions are equivalent: i) the carré du champ Γ[a] is absolutely continuous with respect to the trace τ and its Radon- Nikodym derivative dΓ[a] dτ is bounded (which we write shortly Γ[a] ∈ M) (∂a|(∂a)b)H = τ(bΓ[a]) b ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) there exists a constant Ca ≥ 0 such that |⟨Γ[a], b⟩| ≤ Caτ(|b|) b ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' iii) the vector ∂a ∈ H is right-τ-bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' there exists a constant Ca ≥ 0 such that ∥∂(a)b∥2 H ≤ Caτ(b∗b), b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The set AE ⊆ B whose elements and their adjoint satisfy the conditions above is a ∗- subalgebra of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' is straightforward and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' is a consequence of the symmetry and the Leibnitz rule of the derivation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' AE will be called the Lipschitz algebra of the Dirichlet space (E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For a ∈ AE, we shall denote R(a) : L2(A, τ) → H the bounded operator characterized by R(a) : L2(A, τ) → H R(a)b := (∂a)b b ∈ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice that due to the Leibnitz rule for ∂, one has, for b ∈ B R(a)b = ∂(a)b = ∂(ab) − a∂b = [∂, a]b so that a belongs to AE if and only if it has bounded commutator with the derivation ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The smooth subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We show that the Lipschitz algebra AE contains a subalgebra of elements in the operator domain of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let DM(L) be the domain of the generator on the von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then the space A2,∞ E := AE ∩ DM(L) is a ∗-subalgebra of the Lipschitz algebra AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Observe first that for t > 0, since (I + tL)−1 is a *-weakly continuous contraction of M and E is symmetric with respect to τ, (I + tL)−1 will be also a contraction of the predual L1(A, τ) = M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice then that B2 ⊂ L1(A, τ) is dense in L1(A, τ): in fact if x ∈ M is orthogonal to B2, one has 0 = τ(bax) = (a∗|xb)L2(A,τ) for all a, b ∈ B so that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Hence B ∩ L1(A, τ) is dense in L1(A, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Consider now a ∈ A2,∞ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' According to [CS], for b ∈ B, one has 2⟨Γ[a], b⟩ = (L1/2(a)|L1/2(ab))L2(A,τ) + (L1/2(ab∗)|L1/2a)L2(A,τ) − (L1/2(a∗a)|L1/2b)L2(A,τ) so that, if Γ[a] ∈ M and L(a) ∈ M, there exists a constant C such that � L1/2(a∗a)|L1/2b)L2(A,τ) �� ≤ C τ(|b|) , b ∈ B ∩ L1(A, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In particular, for any t > 0 and b ∈ B ∩ L1(A, τ), b ≥ 0 : ⟨L( I I + tL(a∗a)), b⟩L2(A,τ) = ⟨L1/2(a∗a), L1/2( I I + tL(b))⟩L2(A,τ) ≤ C τ( I I + tL(b)) ≤ Cτ(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This estimate implies that ��L((I + tL)−1(a∗a)) �� M ≤ C is bounded uniformly on t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As lim t↓0 (I +tL)−1(a∗a) = a∗a, σ-weakly in M, and L is a σ-weakly closed operator on M, we have proved a∗a ∈ DM(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Examples by proper, conditionally negative type functions on discrete groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let G be a discrete group with the Haagerup property and ℓ a proper, conditionally negative type function on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The operator L of multiplication by ℓ in l2(G) = L2(C∗ red(G), τ) is the generator of the Dirichlet form E : l2(G) → [0, +∞] E[a] = � g∈G |a(g)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' (τ being the canonical trace on the reduced C∗-algebra of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let λ be the left regular representation of A = C∗ red(G) and define A∞ as the algebra of elements a = � g∈G a(g)λ(g) in C∗ red(G) whose sequence of Fourier coefficients has finite support ({g ∈ G , a(g) ̸= 0} is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' It is known that there exists a unitary representation π of G in on a Hilbert space h and a 1-cocycle c : G → h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' c(gg′) = c(g) + π(g)c(g′) such that the conditionally negative type definite function ℓ can be represented as ⟨c(g′), c(g)⟩h = 1 2 � ℓ(g) + ℓ(g′) − ℓ(g′−1g) � , ℓ(g) = ∥c(g)∥2 , g, g′ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The tangent bimodule is then H = h ⊗ l2(G) where G acts on the left by the diagonal representation π ⊗λ and acts on the right by 1h ⊗ρ where ρ is the right action of G on ℓ2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For a ∈ A∞, the derivation representing E is given by ∂a = � G a(g)c(g) ⊗ δg , a ∈ A∞ 14 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT and the carré du champ is indeed an element of A∞ Γ[a] = � g1,g2∈G a(g2) a(g1)⟨c(g2), c(g1)⟩hλ(g−1 2 g1) = 1 2 � g1,g2∈G a(g2) a(g1) � ℓ(g2) + ℓ(g1) − ℓ(g−1 2 g1) � λ(g−1 2 g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' So that A∞ ⊂ A2,∞ L ⊂ AL, which implies that both AL and A2,∞ L are dense subalgebras of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Notice that Γ[λ(g)] = ℓ(g) Iℓ∞(G) is an invertible element of A∞ whenever ℓ(g) ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' for elements of G outside a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As a consequence, as < b, Γ[a]b >= ||R(a)b||2, one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) R(λ(g))∗R(λ(g)) = ℓ(g) Iℓ∞(G) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The spectral triple of a Dirichlet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' From now on and till the end of this paper, we assume the generator L of the Dirichlet form E to have discrete spectrum away from its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us consider the triple (L2(A, τ) ⊕ H , AE , D) where the Dirichlet algebra AE, as a subalgebra of A, acts on L2(A, τ) ⊕ H by the diagonal action of A on the left, both on L2(A, τ) and H and the Dirac operator is defined as D = � 0 ∂∗ ∂ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The triple (L2(A, τ)⊕H , AE , D) is an essentially discrete spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In particular i) the commutator of the derivation ∂ with the actions of AE on L2(A, τ) and H is given by [∂, a ] = R(a) for all a ∈ AE with R(a)b = (∂a)b for all b ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) for a ∈ AE we have ∥[D, a]∥ = max(∥R(a)∥, ∥R(a∗)∥) and � D, a � = � 0 [∂∗, a] [∂, a] 0 � = � 0 −R(a∗)∗ R(a) 0 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' iii) in the polar decomposition ∂ = u L1/2 of the derivation ∂, the partial isometry u : L2(A, τ) → H is such that u∗u = IL2(Aτ) − p0 and uu∗ = IH − q0, where p0 and q0 are the orthogonal projections onto ker(∂) = ker(L) and ker(∂∗) = Im(∂)⊥, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' iv) one has D2 = � L 0 0 uLu∗ � and |D| = � L1/2 0 0 uL1/2u∗ � = � u∗∂ 0 0 ∂u∗ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' v) if we enumerate λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · the nonzero eigenvalues of L , the corresponding enumeration for |D| is √ λ1 ≤ √ λ1 ≤ √ λ2 ≤ √ λ2 ≤ · · · ≤ √ λn ≤ √ λn ≤ · · · i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' λn(|D|) = λ1/2 [(n+1)/2] and µn(|D|−1) = λ−1/2 [n/2]+1 ([r] being the integer part of a real r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' vi) the projection onto the kernel of D is P0 = � p0 0 0 q0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Straightforward by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ On compact quantum groups, spectral triples of the above type has been constructed in [CFK] Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4, staring from the Dirichlet form of GNS-symmetric noncommutative Levy processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS15 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In all interesting examples, ker(∂∗) is infinite dimensional and then, even if L has a finite dimensional kernel (which often occurs), P0 may have, in general, infinite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The Fredholm modules of a Dirichlet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' According to subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 and with the notations of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8, the Fredholm operators associated to the Dirichlet spaec are F0 = P0 + D |D| = � p0 u∗ u q0 � and F = P0 + D √ 1 + D2 = � p0 I √I+L∂∗ ∂ I √I+L q0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' They differ by an element in the ideal generated by |D|−2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6 and point v) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8 lead to the following representation Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' i) For a = a∗ ∈ AE there exist bounded operators α, β and γ such that i[F, a] = α|D|−1 + |D|−1β + |D|−1/2γ |D|−1/2 with ||α|| ≤ 2 || [D, a] = 2||R(a)||, ||β|| ≤ 2 ||R(a)||, ||γ|| ≤ || R(a) ||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) µ8k([F, a]) ≤ 5 max(||R(a)|| ||R(a∗||) λ−1/2 k+1 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' When the Lipschitz algebra is not dense in A, as in the case of harmonic forms on the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' fractals where AE reduces to constant functions, an alternative, natural choice for the Fredholm module is (H, A, �F) where �F = q⊥ 0 − q0 is the orthogonal symmetry on H with respect to the subspace of gradients Im(∂) = q0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This is what has been done in [CS2] for post critically finite fractals].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This alternative choice leads to different estimates for the quantum derivative (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [CS2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' However, up to a sign, they lead to the same K-homology class, as shown in the following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' If L has discrete spectrum and AE is dense in A, the Fredholm modules (L2(A, τ) ⊕ H, A, F) and � L2(A, τ) ⊕ H, A, I ⊕ (− �F) � are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We forget about p0 which is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' (L2(A, τ) ⊕ H, A, F) and (L2(A, τ) ⊕ H, A, F0) are obviously homotopic (through the family (L2(A, τ) ⊕ H, A, (1 − t)F + t F0), t ∈ [0, 1], while (L2(A, τ)⊕H, A, F0) and � L2(A, τ)⊕H, A, I ⊕(− �F) � are homotopic through the family of Fredholm operators � sin θ I cos θ u∗ cos θu (1 + sin θ)q0 − sin θ I � θ ∈ [0, π/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Commutators with elements of the smooth subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In this subsection, we start with some selfadjoint element a ∈ A2,∞ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The goal is to prove that the commutator [ |D|, a ] is bounded, so that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We start with some intermediary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [L, a] 1 √ 1 + L is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us compute, for c ∈ A and b ∈ DomL2(L) ∩ B and making use of the rules established in [CS, square roots] and making use of the symmetry of the A-A-bimodule H : (c|[L, a]b) = τ(c∗(L(ab) − aL(b))) = τ(c∗(−2Γ(a∗, b) + L(a)b) = −2(∂(a∗)c, ∂b) + (c, L(a)b) = −2(c, R(a∗)∗∂b) + (c, L(a)b) 16 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT from which we deduce (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4) [L, a] = −2R(a∗)∗∂ + L(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ As a consequence, we establish the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For γ ∈ (0, 1/2), (I + L)1/2−γ [(I + L)γ, a ] is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As a consequence, [(I + L)γ, a ] is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Preuve : We start with � (1 + L)γ , a � = Cγ � +∞ 0 tγ−1 � 1 + L t + 1 + L , a � dt = −Cγ � +∞ 0 tγ−1 � t t + 1 + L , a � dt = Cγ � +∞ 0 tγ 1 t + 1 + L [L, a] 1 t + 1 + L dt = Cγ � +∞ 0 tγ 1 t + 1 + L [L, a] 1 √ 1 + L √ 1 + L t + 1 + L dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Coupling with x, y ∈ L2(A, τ), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='13 provides some constant C′ such that : ��� � y, � (1 + L)γ , a � x ���� ≤ C′ � +∞ 0 tγ ��� 1 t + 1 + L y ��� ��� √ 1 + L t + 1 + L x ��� dt ≤ C′ �� +∞ 0 t2γ� y , 1 (t + 1 + L)2 y � dt �1/2 �� +∞ 0 � x , 1 + L (t + 1 + L)2 x � dt �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' One checks easily that � +∞ 0 t2γ 1 (t + 1 + L)2dt is proportional to (1 + L)2γ−1, and that � +∞ 0 1 + L (t + 1 + L)2dt is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' From which ��� � y, � (1 + L)γ , a � x ���� ≤ C′′||(1 + L)γ−1/2y || ||x|| and the result □ As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' a corollary, we get Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [ √ 1 + L , a ] is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='14 with γ = 1/4 : (I + L)1/4[(I + L)1/4, a] and [(I + L)1/4, a](I + L)1/4 are bounded operators (for the latter, consider the adjoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The Leibnitz rule provides [(1 + L)1/2, a] = (1 + L)1/4 [ (1 + L)1/4, a ] + [(1 + L)1/4, a ] (1 + L)1/4 which is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ We can now prove the main result of this section : Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For a ∈ A2,∞ L the operator [ |D|, a] is a bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Hence, the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='11 hold true for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let ∂ = uL1/2 be the polar decomposition of ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As [∂, a] = [u, a| L1/2+u [a, L1/2] and u [a, L1/2] are bounded, [u, a| L1/2 is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We have now � (∂∂∗)1/2, a � = � uL1/2u∗, a � = [∂, a]u∗ + uL1/2 [u∗, a] = [∂, a]u∗ + u � [a∗, u]L1/2�∗ which is a bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lower bounds on singular values of quantum differentials In this section we consider conditions providing lower bounds for the singular values µk(i[F, a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We also propose general and particular examples where these conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lower bounds on quantum differentials of Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Suppose that a ∈ A2,∞ L satisfies the three conditions : i) The commutator [ √ I + L, a] is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ii) R(a)∗R(a) has a finite dimensional kernel iii) R(a)∗R(a) is invertible on ker(R(a))⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then there exists an integer k0 and a constant C(a) such that µk(i[F, a]) ≥ C(a) (1 + ε(k)) λk+k0(L)−1/2 where C(a) is the infimum of the spectrum of |R(a)| on ker(R(a))⊥, ε(k) is a sequence tending to 0 as k → ∞ and k0 = dim(ker(L)) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us start with an observation : as � 0 0 [∂ I √I+L, a] 0 � = � 0 0 I 0 � � p0 [ I √I+L∂∗, a] [∂ I √I+L, a] q0 � � 0 0 0 I � which provides µk([F, a]) ≥ µk � [∂ I √ I + L2, a] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We have then [∂ I √ I + L, a] = [∂, a] I √ I + L − ∂ I √ I + L �√ I + L , a � I √ I + L i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=', since [∂, a] is equal to R(a), ∂ I √ I + L is bounded and �√ I + L , a � is compact, we get [∂ I √ I + L, a] = � R(a) + κ � I √ I + L with κ compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='15, the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ In the sequel, we show that the conditions of the above result are realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' In particular, according to equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3 in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7, condition ii) is satisfied whenever E is the Dirichlet form associated with a proper negative type function ℓ on a discrete group G and a = λ(g), g ∈ G, ℓ(g) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 18 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Roots of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ([CS1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Suppose that �E is a (symmetric, regular, completely) Dirichlet form on L2(A, τ) with generator �L such that A2,∞ �L is dense in A, and consequently A�L is dense in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Fix any β ∈ (0, 1), let E be the Dirichlet form with generator L = �Lβ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [CS1] Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The semigroup e−tL is called subordinated to the semigroup e−t�L (see [C1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We have A2,∞ �L ⊂ A2,∞ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We start with the standard formula for roots of positive operators : �Lβ = Cβ � +∞ 0 tβ−1 �L t + �L dt = Cβ � +∞ 0 s−β �L I + s�L ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1) with Cβ = sin(βπ)/βπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As I I + s�L acts as a completely positive contraction of M, for a ∈ A2,∞ �L we have on one hand �� �L I + s�L (a) �� M| ≤ ∥�L(a)∥M so that the integral converges in M for s → 0, and on the other hand �� �L I + s�L (a) �� M = 1 s ��a − I I + s�L (a) �� M ≤ 2 s||a||M so that the integral converges in M as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We have proved A2,∞ �L ⊂ DM(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As A2,∞ �L is an involutive algebra, for a ∈ A2,∞ �L we have also a∗ ∈ DM(L) and a∗a ∈ DM(L), so that Γ[a] = 1 2 � a∗L(a) + L(a)∗a − L(a∗a) � ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Which proves A2,∞ �L ⊂ AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' With L = �Lβ as in formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1), there exists an involutive subalgebra A∞ of A2,∞ L dense in A such that, for any a ∈ A∞, the commutator [ √ I + L, a] is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Take A∞ = A2,∞ �L and apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='14 with γ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Slow conditionally negative type function on discrete groups In this section we come back to the framework of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Thus, G is a discrete group with the Haagerup property, τ is the canonical trace on C∗ red(G), ℓ is a proper negative type function on G, L is the operator of multiplication by ℓ in ℓ2(G) = L2(C∗ red(G), τ) and Eℓ is the Dirichlet form with generator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We recall that A∞ is the dense ∗-subalgebra of A = C∗ red(G) of the a = � g∈G a(g)λ(g) with finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Asymptotic orthogonality for 1-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let us start with a simple observation: for fixed g ∈ G we have ℓ(g−1g′) = ℓ(g) + ℓ(g′) − 2(c(g)|c(g′)) = ℓ(g′) + O(ℓ(g′))1/2 as g′ → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We are going to show that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 applies to the Dirichlet forms Eℓ provided the negative type function ℓ satisfies a strengthened form of the above asymptotic estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' (Slow conditionally negative type functions) A conditionally negative type function ℓ : G → [0, +∞) is said to be slow if it is proper and if, for any fixed g ∈ G, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1) ℓ(g−1g′) = ℓ(g′) + o(ℓ(g′))1/2 as g′ → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' QUANTUM DIFFERENTIALS OF SPECTRAL TRIPLES, DIRICHLET SPACES AND DISCRETE GROUPS19 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let ℓ : G → 0, +∞) be slow conditionally negative type function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 applies to the Dirichlet form Eℓ for any a = λ(g) with ℓ(g) ̸= 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' all g ∈ G but a finite number), so that there exists a sequence ε(k) → 0 such that � ℓ(g) (1 + ε(k)) λk+1(L)−1/2 ≤ µk � i � F, λ(g) �� ≤ 5 � ℓ(g) λ[k/8]+1(L)−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' One checks easily that for such a = λ(g), [ √ I + L, a] = kgλ(g) where kg is the multi- plication operator by the function g′ → � 1 + l(g−1g′) − � 1 + ℓ(g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Check that � 1 + l(g−1g′) − � 1 + ℓ(g′) = ℓ(g−1g′) − ℓ(g′) � 1 + l(g−1g′) + � 1 + ℓ(g′) = o(ℓ(g′)1/2) � 1 + l(g−1g′) + � 1 + ℓ(g′) tends to 0 as g′ → ∞, so that kg is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Condition (i) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Condition (ii) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 is provided by identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The estimates come from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6, Conclusion 5 of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='8 and identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3) of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let �ℓ be a proper conditionally negative type function on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then ℓ := �ℓβ is a slow conditionally negative type function, for an arbitrary β ∈ (0, 1) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 applies to the Dirichlet form Eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Fix g and make g′ tend to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As noticed above, one has �ℓ(g−1g′) = �ℓ(g′)+O(�ℓ(g′)1/2) = �ℓ(g′) � 1 + O(�ℓ(g′)−1/2� so that ℓ(g−1g′)) = �ℓ(g−1g′)β = �ℓ(g′)β(1 + O(�ℓ(g′)−1/2) = ℓ(g′) + o(ℓ(g′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Suppose that G = Fp is the free group with p generators and ℓ is the length function, which is conditionally negative (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' [Haa]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='1 applies to the Dirichlet form Eℓ for a = λ(g) with g ̸= e so that there exist constants C1, C2 > 0 and an integer k0 such that C1(Log(k))−1/2 ≤ µk(i[F, a]) ≤ C2(Log(k))−1/2 , k ≥ k0 with C1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' C2) arbitrarily close to � ℓ(g) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 5 � ℓ(g)) if K0 is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The first assumption is straightforward: since ℓ is a length function, we have |ℓ(s−1t)− ℓ(t)| ≤ ℓ(s) so that ℓ is a proper, slow, conditionally negative type function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' For the second assumption, check that the ball Bm of radius m in G, Bm = {g ∈ G | ℓ(g) ≤ m}, has a cardinality |Bm| = 1 + p p − 1(2p − 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' As λk = m for |Bk| ≤ m < ∥Bk+1|, we get λk ∼ k Log(2p − 1), k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2 provide the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Weight functions as slow conditionally negative type functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Here we prove that all weight functions on discrete groups are slow negative type functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Let ℓ be a conditionally negative type function on the group G (symmetric and strictly positive away from the unit) and let (hℓ, πℓ, c) be the associated 1-cocycle c : G → hℓ with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='2) c(st) = c(s) + πℓ(s)c(t) , ||c(t)||2 = ℓ(t) , (c(s)|c(t))H = 1 2 � ℓ(s) + ℓ(t) − ℓ(s−1t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' 20 FABIO E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' CIPRIANI, JEAN-LUC SAUVAGEOT One checks easily that the vectors c(s), c(t) in hℓ are orthogonal if and only if e lies between s and t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' ℓ(s−1t) = ℓ(s) + ℓ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The function √ ℓ is conditionally negative too and provides a left-invariant metric on G by d√ ℓ(s, t) := � ℓ(s−1t) s, t ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The cocycle is an isometric embedding of the metric space (G, d√ ℓ) into the Hilbert space hℓ ∥c(t) − c(s)∥hℓ = ∥c(ss−1t) − c(s)∥hℓ = ∥c(s) + πℓ(s)(c(s−1t)) − c(s)∥hℓ = ∥c(s−1t))∥hℓ = � ℓ(s−1t) = d√ ℓ(s, t) s, t ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The characteristic property of a slow negative type function, expressed as ℓ(s) + ℓ(t) − ℓ(s−1t) = o( � ℓ(t)) t → ∞, for each fixed s ∈ G, can be restated as 0 = lim t→∞ ℓ(s) + ℓ(t) − ℓ(s−1t) 2 � ℓ(s) � ℓ(t) = lim t→∞ (c(s)|c(t))hℓ ∥c(s)∥Hℓ · ∥c(t)∥hℓ , s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' The property thus refers to the asymptotic orthogonality of t ∈ G with respect to any fixed s ∈ G, when these are embedded in the Hilbert space hℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' We recall that a weight on a discrete group G ([deH]) is a negative type function such that ℓ : G → [0, +∞) ℓ(st) ≤ ℓ(s) + ℓ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Any proper weight function on a discrete group G is a slow, conditionally negative negative type function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Since a weight satisfies |ℓ(s) − ℓ(t)| ≤ ℓ(s−1t), we have 0 ≤ ℓ(s) + ℓ(t) − ℓ(s−1t) ≤ 2(ℓ(s) ∧ ℓ(t)) and ℓ(s) + ℓ(t) − ℓ(s−1t) 2 � ℓ(s) � ℓ(t) ≤ ℓ(s) ∧ ℓ(t) � ℓ(s) � ℓ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Since ℓ is proper we have lim t→∞ ℓ(s) + ℓ(t) − ℓ(s−1t) 2 � ℓ(s) � ℓ(t) ≤ lim t→∞ ℓ(s) ∧ ℓ(t) � ℓ(s) � ℓ(t) = lim t→∞ � ℓ(s) ℓ(t) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' A weight function gives rise to a left-invariant metric on G: dℓ(s, t) := ℓ(s−1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' If k := inf{ℓ(t) : t ∈ G, t ̸= e}, the cocycle is a Lipschitz embedding of the metric space (G, dℓ) into the real Hilbert space hℓ ∥c(t) − c(s)∥H = � ℓ(s−1t) ≤ � k−1ℓ2(s−1t) = 1 √ k ℓ(s−1t) = 1 √ k dℓ(s, t) s, t ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' Examples include length functions of free groups Fn and the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content=' REFERENCES [AHK] S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='it (JLS) Institut de Mathématiques de Jussieu – Paris Rive Gauche, CNRS – Université Paris Cité, F-75205 Paris Cedex 13, France Email address: jean-luc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='sauvageot@imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQfVPdi/content/2301.00140v1.pdf'} diff --git a/RNE3T4oBgHgl3EQfDAny/content/tmp_files/2301.04283v1.pdf.txt b/RNE3T4oBgHgl3EQfDAny/content/tmp_files/2301.04283v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a437eb29d997fb1cc89dd0acff4b9007775b8f5 --- /dev/null +++ b/RNE3T4oBgHgl3EQfDAny/content/tmp_files/2301.04283v1.pdf.txt @@ -0,0 +1,1385 @@ +A Multi-Modal Geographic Pre-Training Method +Ruixue Ding†∗, Boli Chen†∗, Pengjun Xie†, Fei Huang†, Xin Li‡, Qiang Zhang‡, Yao Xu‡ +†DAMO Academy, Alibaba Group +‡Gaode Map, Alibaba Group +{ada.drx,boli.cbl,chengchen.xpj,f.huang,beilai.bl,muxi.zq,xuenuo.xy}@alibaba-inc.com +ABSTRACT +As a core task in location-based services (LBS) (e.g., navigation +maps), query and point of interest (POI) matching connects users’ in- +tent with real-world geographic information. Recently, pre-trained +models (PTMs) have made advancements in many natural lan- +guage processing (NLP) tasks. Generic text-based PTMs do not +have enough geographic knowledge for query-POI matching. To +overcome this limitation, related literature attempts to employ +domain-adaptive pre-training based on geo-related corpus. How- +ever, a query generally contains mentions of multiple geographic +objects, such as nearby roads and regions of interest (ROIs). The +geographic context (GC), i.e., these diverse geographic objects and +their relationships, is therefore pivotal to retrieving the most rele- +vant POI. Single-modal PTMs can barely make use of the important +GC and therefore have limited performance. In this work, we pro- +pose a novel query-POI matching method Multi-modal Geographic +language model (MGeo), which comprises a geographic encoder +and a multi-modal interaction module. MGeo represents GC as a +new modality and is able to fully extract multi-modal correlations +for accurate query-POI matching. Besides, there is no publicly avail- +able benchmark for this topic. In order to facilitate further research, +we build a new open-source large-scale benchmark Geographic +TExtual Similarity (GeoTES). The POIs come from an open-source +geographic information system (GIS). The queries are manually +generated by annotators to prevent privacy issues. Compared with +several strong baselines, the extensive experiment results and de- +tailed ablation analyses on GeoTES demonstrate that our proposed +multi-modal pre-training method can significantly improve the +query-POI matching capability of generic PTMs, even when the +queries’ GC is not provided. Our code and dataset are publicly +available at https://github.com/PhantomGrapes/MGeo. +CCS CONCEPTS +• Information systems → Language models; Similarity mea- +sures; Business intelligence. +KEYWORDS +query-POI matching, multi-modal, language model, geographic +context, benchmark +∗Equal contribution. +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +Preprint, +© 2023 Copyright held by the owner/author(s). +https://doi.org/10.1145/nnnnnnn.nnnnnnn +Figure 1: A typical query-POI matching procedure. +Reference Format: +Ruixue Ding, Boli Chen, Pengjun Xie, Fei Huang, Xin Li, Qiang Zhang, Yao +Xu. 2023. A Multi-Modal Geographic Pre-Training Method. Preprint. 10 +pages. +1 +INTRODUCTION +As an essential function of location-based services (LBS) like naviga- +tion maps (e.g., Google Maps), ride-hailing applications (e.g., Uber), +and food delivery platforms (e.g., Uber Eats), query and point of +interest (POI) matching aims to find a list of candidate POIs based +on users’ specific or implicit intent. Since the candidate results +are crucial for providing real-world geographic information to the +users, which directly impacts the navigation, routing, and ordering +process, effective and accurate query-POI matching is indispensable +for delivering a satisfactory user experience. A typical query-POI +matching procedure is illustrated in Figure 1, which consists of a +two-stage retrieve-then-rank pipeline [37, 39]. In specific, given a +query, the lightweight retriever first produces an initial set of can- +didate POIs by searching a massive database, then the ranker sorts +the most relevant candidate. This kind of architecture is widely +adopted in information retrieval (IR) systems on account of the +efficiency-effectiveness trade-off. +Recent literature on natural language processing (NLP) as well as +IR shows a flourishing advancement of pre-trained models (PTMs), +arXiv:2301.04283v1 [cs.CL] 11 Jan 2023 + +重庆南开 +日 +中学 +Chongqing Nankai +Secondary School +重庆市名 +校联合中 +学 +- +Chongqing Unite +Secondary Scho +- +三峡广场 +Sanxia Square +School gate on underground road +Retrieval +RankingPreprint, +Ding et al. +notably in semantic textual similarity (STS) and open-domain ques- +tion answering (QA) [3, 14, 17]. Since continued self-supervised +training on domain-specific corpus is shown to be effective for +PTMs [9], various domain-adaptation methods have lately been +proposed to inject the geographic knowledge based on geo-related +textual data and relevant user behavioral data [10, 11, 20, 30]. Al- +though better at capturing the semantic similarity than the generic +PTMs, these methods can barely make use of the more important +circumstantial geographic context (GC), i.e., the diverse geographic +objects and their relationships from the geographic information +system (GIS) detailed in Definition 2. Specifically, the geographic +objects consist of roads represented as lines and regions of inter- +est (ROIs) represented as polygons, the relationship includes near, +covered, and their relative position. +As the query usually mentions multiple geographic objects in +the background, extracting the correlations between these objects +is necessary for accurate query-POI matching. For example, given +the query "school gate on underground road", as shown in Figure 1, +several relevant POIs are retrieved. The nearest "underground road" +to the user is the "Nankai Underground Rd", and the "Nankai Sec- +ondary School" has a gate (c) on the "Underground Rd". Therefore, +the most matched POI should be the gate (c). The problem is that the +"Nankai Secondary School" is formally located on the "Shapingba S +St" with its main gate (a). Its side gate (c) is not recorded in the GIS +as located on the "Underground Rd". It should also be noticed that +the user is currently in the "Sanxia Square", which has a gate (b) +located on the "Underground Rd". The semantic textual similarity +alone is not enough to distinguish these two hard negatives (a) +and (b). Moreover, the gate (d) of the "United Secondary School" is +the closest school gate to the user. Simply considering the relative +position of the user and the POI will match the wrong gate (d). Only +by taking the entire GC into consideration can we find the correct +gate (c). +To this end, we propose a novel method Multi-modal Geographic +language model (MGeo), which bridges the modal gap between se- +mantics and GC. MGeo consists mainly of a geographic encoder +and a multi-modal interaction module. The geographic encoder +makes use of the GC by representing it as a new modality. The +multi-modal interaction module then incorporates the geographic +features with the semantics. MGeo makes use of the textual, geo- +graphic, and cross-modal interactions between queries and POIs. +Since the interaction module is compatible with queries that have +no GC, it is optional to provide the queries’ geolocation as many +applications may require. As a result, rich correlations among tex- +tual and geographic modalities can be fully extracted to ensure the +quality of query-POI matching. +In addition, there exists no public unencrypted benchmark for +query-POI matching mostly due to privacy issues. Large publicly +available corpus could lead to many breakthroughs in research, e.g., +MS MARCO [22]. Intending to facilitate further research on this +topic, develop robust techniques, and track progress, we introduce +Geographic TExtual Similarity (GeoTES), which is an open-source +large-scale benchmark for query-POI matching with GC. The POIs +come from the open-source GIS OpenStreetMap (OSM)1. To prevent +1https://www.openstreetmap.org +privacy issues, the queries are manually generated by annotators +thus do not require encryption. +Our major contributions are highlighted as follows: +• We formalize the important concept GC for the query-POI +matching problem and propose a novel method MGeo that +uses geographic encoder to represent it as a new modality. +• We use multi-modal interaction module to incorporate the +correlations among textual and geographic modalities. Be- +sides, MGeo is compatible with queries that have no GC. +• We develop a new open-source large-scale benchmark named +GeoTES. The POIs come from an open-source GIS and the +queries are manually generated by annotators to prevent +privacy issues. +• Compared with strong baselines, the experiment results +demonstrate that our proposed MGeo can significantly im- +prove the query-POI matching capability of generic PTMs, +even when no GC is provided for the queries. +2 +RELATED WORK +2.1 +Relevance Model +Traditional approaches for retrieving documents from large corpus +generally use exact term-level matching, e.g., Okapi Best Match- +ing (BM25) [27]. Despite such heuristic retrievers have low latency +via inverted list data structure, their measurement of similarity is +only based on document statistics. On the other hand, the latent +models can correlate semantically similar terms and reduce the +matching dimensionality, e.g., Partial Least Square (PLS) [28]. Lat- +terly, Deep neural network (DNN) models have been introduced +to IR. For example, Deep Structured Semantic Model (DSSM) [12] +measures the relevance of queries and documents in a semantic +vector space by computing their cosine similarity. Along with the +success of PTMs in NLP, studies on IR have also made remark- +able progress [7, 14, 17]. On account of the efficiency-effectiveness +trade-off, there are two major architectures, i.e., bi-encoder and +cross-encoder [32]. Bi-encoder allows efficient indexing [4, 26] and +is usually used in the retrieval system. In contrast, cross-encoder +concatenates the query and document to perform cross-interaction +over all input terms. Although cross-encoder can provide a more ac- +curate estimation of relevance, it needs more computing resources +and is usually used only in the ranking system. MGeo can use either +the bi-encoder or cross-encoder architecture. +2.2 +Multi-Modal Representation Learning +Following the tremendous success of various pre-training tech- +niques in NLP, a lot of Transformer-based models are proposed for +other modalities, such as compute vision (CV) [1, 6, 16]. Except for +single-modal, recent studies also show the derivative models have +great potential in multi-modal representation learning [2, 16, 18, 29]. +For example, CLIP [25] converts classification to a retrieval task and +enables zero-shot learning via large-scale multi-modal pre-training. +In addition to image, layout of document and table can also be +represented as different modalities [33, 36]. In this paper, we pro- +pose a novel multi-modal geographic pre-training method, which +represents the GC as a new modality for query-POI matching. + +A Multi-Modal Geographic Pre-Training Method +Preprint, +2.3 +Query-POI Matching +Previous work focuses on modeling the relative position between +queries and POIs. Based on DSSM, PALM [39] obtains the positional +relationship of queries and POIs from coordinate-based and kernel- +based location embeddings, and incorporates the relationship with +semantic similarity for POI retrieval. STDGAT [38] further takes +multiple spatiotemporal factors into consideration via dual graph at- +tention network when quantifying the query-POI relevance. On ac- +count of the ubiquity of PTMs in NLP, domain-adaptive pre-training +methods have been proposed to inject extralinguistic knowledge +into the generic PTMs [10, 20]. Typically, GeoL [11] makes use of +the static geographic knowledge based on user behavioral (search +logs), e.g., geocoding [8]. Despite the domain-adapted PTMs may be +better at capturing the semantic similarity than the generic PTMs, +they are still limited by ignoring the GC in the background. +In addition, since the public benchmark can facilitate further +research and play an important role in the development of robust +techniques, we also establish a reliable large-scale query-POI match- +ing benchmark GeoTES. +3 +PRELIMINARY +We first introduce the formal description of the query-POI matching +problem, as well as some important definitions related to GC. Table 1 +gives the frequently used notations. +Let 𝑃 be the set of POIs𝑝. 𝑃 can either contain dozens of candidate +POIs or a large number of POIs in the massive database. Each +POI 𝑝 consists of a textual description 𝑡𝑝 and its geolocation 𝑙𝑝. +The textual description of the POI 𝑡𝑝 contains its formal address +and name. Let 𝑞 denote a query made by the user. The textual +description of the query 𝑡𝑞 belongs to three types, i.e., common +address description, formal street number description, and casual +colloquial description. The street number query contains standard +numerical designation for a target POI, while the address query does +not. The colloquial query uses spoken language and may contain +colloquial noises. +The query’s geolocation 𝑙𝑞 can be the users’ geolocation. When +the user searches for another area using the map, 𝑙𝑞 is the center +location displayed on screen. Furthermore, 𝑙𝑞 may or may not be +provided. We denote geolocation of a POI or query as 𝑙𝑝𝑞. +Problem 1. Query-POI matching problem. Given the POI set +𝑃 and a user’s query 𝑞 in LBS, we aim to estimate the POI 𝑝 ∈ 𝑃 that +best matches the user’s intent. +We define two tasks based on the size of 𝑃, i.e., ranking and +retrieval. Specifically, for the ranking task, 𝑃 is a list of candidate +POIs, where the best-matched one is included. As for the retrieval +task, 𝑃 is the massive database that contains all POIs. Since cross- +encoder is inefficient for large size of 𝑃, it only runs on the ranking +task. Bi-encoder can run on both the ranking and retrieval tasks. +Definition 1. Geographic object. GIS is constructed on spatial +data that defines the real-world geometric space. Let G be the spatial +database. Each geographic object 𝑜 ∈ G with 𝑚 vertices is described +as a sequence of geolocation {𝑙𝑜 +1,𝑙𝑜 +2, . . . ,𝑙𝑜𝑚} and characterized by its +shape 𝑜𝑠 ∈ {𝐿𝐼𝑁𝐸, 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 }. Specifically, 𝐿𝐼𝑁𝐸 represents the +real-world road and 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 represents the ROI. +Table 1: Table of notations. +Notation +Description +𝑃, 𝑝 +The POI set and the POI. +𝑞 +The query of user. +𝑜 +The geographic object. +𝑜𝑠 +The shape of geographic object, ∈ {𝐿𝐼𝑁𝐸, 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 }. +𝑜𝑚 +The position of 𝑜 in the map. +𝑡 +The textual description of POI or query. +𝑙 = (𝑙𝑛𝑔,𝑙𝑎𝑡) +The geolocation represented by longitude and latitude. +𝑙𝑝𝑞 +The geolocation of a POI or query. +𝑙𝑜 +A vertex of 𝑜. +˜𝑜 +The rectangle that approximates the shape of 𝑜. +𝑟𝑡 +The relation type ∈ {𝑁𝐸𝐴𝑅,𝐶𝑂𝑉 𝐸𝑅𝐸𝐷 }. +𝑟𝑝 +The relative position. +Here we use 𝑚 to denote the number of vertices in 𝑜. Note that +given the geolocation of the POI or the query, we can form a list of +nearby geographic objects {𝑜1,𝑜2, . . . ,𝑜𝑛} sorted by distance, i.e., +𝑜1 is the nearest geographic object to the POI or query. 𝑛 is used to +denote the number of geographic objects for a geolocation 𝑙𝑝𝑞. We +export OSM to PostGIS2 and get the Geographic Context (GC) of a +geolocation from it. +Definition 2. Geographic Context (GC). Given the geolocation +𝑙𝑝𝑞 of a POI or query, where 𝑙𝑝𝑞 is represented by a geographic coor- +dinate (𝑙𝑛𝑔,𝑙𝑎𝑡), the GC is characterized by the relationships between +𝑙𝑝𝑞 and its 𝑛 geographic objects {𝑜1,𝑜2, . . . ,𝑜𝑛}. Formally, the rela- +tion type 𝑟𝑡 ∈ {𝑁𝐸𝐴𝑅,𝐶𝑂𝑉𝐸𝑅𝐸𝐷} indicates whether 𝑙𝑝𝑞 is inside +𝑜𝑖 or at a distance. The relative position 𝑟𝑝 depicts a more detailed +positional relationship between 𝑙𝑝𝑞 and 𝑜𝑖. +When searching for a target POI, a user usually explores nearby +circumstantial spatial data and mentions multiple related geographic +objects in the query. Besides, the intrinsic characteristics of geo- +graphic objects are also important GC features (described in Sec- +tion 4.1.1). Therefore, the GC is pivotal to ensuring the quality of +query-POI matching. +4 +METHOD +In this section, we present the detailed architecture and pre-training +process of MGeo. Following state-of-the-art multi-modal meth- +ods [2, 16, 19], MGeo is composed of a geographic encoder and +a multi-modal interaction module, as shown in Figure 2. The full +training process of MGeo consists of three steps. First, we train ge- +ographic encoder alone to learn representations of GC. The trained +geographic encoder is fixed in the following stages. Then, text- +geolocation pairs are used to pre-train MGeo in a multi-modal way. +By modeling geographic objects along with text and pre-training +with massive text-geolocation pairs, MGeo successfully aligns these +two modals into a same latent space. Lastly, MGeo is fine-tuned on +ranking and retrieval tasks and gains significant improvements. +4.1 +Geographic Encoder +The geolocation alone is meaningless unless it has GC. Taking a +geolocation 𝑙 as input, geographic encoder maps the GC as a new +modality to dense representations, which contains features of the +surrounding geographic objects {𝑜1,𝑜2, . . . ,𝑜𝑛}. +2https://postgis.net/ + +Preprint, +Ding et al. +Figure 2: Architecture of MGeo. Left part shows encoding and pre-training process of geographic encoder and right part shows +the multi-modal pre-training process of MGeo. Word embeddings of text t and GC representations of geographic encoder h +are concatenated together and fed to multi-modal interaction module, which produces final representations ˆh𝑡 for each text +token and ˆh𝑙 for each geographic object. +4.1.1 +Encoding. Geographic encoder can extract the relationships +between geolocation and geographic objects. For a geographic ob- +ject 𝑜𝑖, a one-hot function is used to encode the categorical relation +type 𝑟𝑡 +𝑖 as a numeric array and to obtain its corresponding embed- +dings e𝑡 +𝑖 . To simplify the relative position 𝑟𝑝 +𝑖 , we form a rectangle +˜𝑜𝑖 of similar size to approximate the shape of 𝑜𝑖. Each side of the +substituted rectangle (left, bottom, right, and top) is defined as: +˜𝑜𝑙𝑒𝑓 𝑡 +𝑖 += +𝑚𝑖𝑛�{𝑙𝑛𝑔𝑜𝑖 +𝑗 }𝑗 ∈{1,...,𝑚𝑖 } +�, +(1) +˜𝑜𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 += +𝑚𝑖𝑛�{𝑙𝑎𝑡𝑜𝑖 +𝑗 }𝑗 ∈{1,...,𝑚𝑖 } +�, +(2) +˜𝑜𝑟𝑖𝑔ℎ𝑡 +𝑖 += +𝑚𝑎𝑥 �{𝑙𝑛𝑔𝑜𝑖 +𝑗 }𝑗 ∈{1,...,𝑚𝑖 } +�, +(3) +˜𝑜𝑡𝑜𝑝 +𝑖 += +𝑚𝑎𝑥 �{𝑙𝑎𝑡𝑜𝑖 +𝑗 }𝑗 ∈{1,...,𝑚𝑖 } +�, +(4) +where 𝑙𝑛𝑔 denotes longitude and 𝑙𝑎𝑡 denotes latitude of 𝑙𝑜𝑖 +𝑗 for sim- +plicity. The relative position 𝑟𝑝 +𝑖 = {𝑟𝑝𝑙𝑒𝑓 𝑡 +𝑖 +,𝑟𝑝𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 +,𝑟𝑝𝑟𝑖𝑔ℎ𝑡 +𝑖 +,𝑟𝑝𝑡𝑜𝑝 +𝑖 +} is +then calculated by the normalized distances between 𝑙𝑝𝑞 and each +side of the ˜𝑜𝑖. For example, 𝑟𝑝𝑙𝑒𝑓 𝑡 +𝑖 +is calculated as: +𝑟𝑝𝑙𝑒𝑓 𝑡 +𝑖 += 𝑠𝑔𝑛(𝑙𝑛𝑔𝑝𝑞 − ˜𝑜𝑙𝑒𝑓 𝑡 +𝑖 +) ∗ 𝑚𝑖𝑛�𝑘, ⌊𝑘 +|𝑙𝑛𝑔𝑝𝑞 − ˜𝑜𝑙𝑒𝑓 𝑡 +𝑖 +| +˜𝑜𝑟𝑖𝑔ℎ𝑡 +𝑖 +− ˜𝑜𝑙𝑒𝑓 𝑡 +𝑖 +⌋� + 𝑘, +(5) +where 𝑠𝑔𝑛(·) is the sign function, and ⌊·⌋ is the floor function that +outputs the greatest integer less than or equal to a number. 𝑘 ∈ +N is a discretization factor that maps the relative distance ratio +to a discrete number. As a result, we have 𝑟𝑝𝑙𝑒𝑓 𝑡 +𝑖 +∈ {0, . . . , 2𝑘}. +The discretized relative position feature is then encoded as e𝑝 +𝑖 = +{e𝑝𝑙𝑒𝑓 𝑡 +𝑖 +, e𝑝𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 +, e𝑝𝑟𝑖𝑔ℎ𝑡 +𝑖 +, e𝑝𝑡𝑜𝑝 +𝑖 +}. +To extract the intrinsic features of geographic objects, the OSM +IDs are mapped to embeddings in a similar way to word embeddings. +The shape type 𝑜𝑠 is also mapped to embeddings like relation type +𝑟𝑡. The ID embeddings of 𝑜𝑖 are denoted as e𝑑 +𝑖 and its shape type +embeddings as e𝑠 +𝑖 . +Furthermore, to recognize the relationships among geographic +objects, such as overlapping, the entire map area as a rectangle is +split into a 𝑁 × 𝑁 grid to obtain its scale factors 𝑠𝑙𝑛𝑔 and 𝑠𝑙𝑎𝑡 for +longitude and latitude respectively: +𝑠𝑙𝑛𝑔 = 𝑙𝑛𝑔𝑚𝑟𝑖𝑔ℎ𝑡 − 𝑙𝑛𝑔𝑚𝑙𝑒𝑓 𝑡 +𝑁 +,𝑠𝑙𝑎𝑡 = 𝑙𝑎𝑡𝑚𝑡𝑜𝑝 − 𝑙𝑎𝑡𝑚𝑏𝑜𝑡𝑡𝑜𝑚 +𝑁 +, +(6) +where 𝑙𝑛𝑔𝑚𝑟𝑖𝑔ℎ𝑡 denotes longitude of the map’s right side and so +on. The position of 𝑜𝑖 in the map 𝑜𝑚 +𝑖 can thus be calculate with the +scale factors. For example, 𝑜𝑚𝑙𝑒𝑓 𝑡 +𝑖 +and 𝑜𝑚𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 +are calculated as: +𝑜𝑚𝑙𝑒𝑓 𝑡 +𝑖 += +⌊ ˜𝑜𝑙𝑒𝑓 𝑡 +𝑖 +−𝑙𝑛𝑔𝑚𝑙𝑒𝑓 𝑡 +𝑠𝑙𝑛𝑔 +⌋ +∈ N, +(7) +𝑜𝑚𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 += +⌊ ˜𝑜𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 +−𝑙𝑎𝑡𝑚𝑏𝑜𝑡𝑡𝑜𝑚 +𝑠𝑙𝑎𝑡 +⌋ +∈ N. +(8) +The discretized position feature of 𝑜𝑖 in the map is then encoded +as e𝑚 +𝑖 += {e𝑚𝑙𝑒𝑓 𝑡 +𝑖 +, e𝑚𝑏𝑜𝑡𝑡𝑜𝑚 +𝑖 +, e𝑚𝑟𝑖𝑔ℎ𝑡 +𝑖 +, e𝑚𝑡𝑜𝑝 +𝑖 +}. Finally, the geographic +encoder sums these features of 𝑜𝑖 up as: +e𝑖 = e𝑡 +𝑖 + e𝑑 +𝑖 + e𝑠 +𝑖 + +∑︁ +e𝑝 +𝑖 + +∑︁ +e𝑚 +𝑖 +(9) +The intrinsic characteristics of geographic objects are described +by the three components (e𝑑, e𝑠, and e𝑚). e𝑑 is the unique identifier +of a geographic object, e𝑠 distinguishes road from AOI, e𝑚 depicts +the positional relation among different geographic objects. The +other two components (e𝑡 and e𝑝) describe correlations between +geolocation and geographic objects. After encoding surrounding +geographic objects as a sequence {e1, . . . , e𝑚}, geographic encoder +employs multi-layer bidirectional transformers [34] to learn inter- +actions among them. Following previous work [32], a 𝐺𝐶 token +is prepended at the beginning like the 𝐶𝐿𝑆 token. The outputs of +geographic encoder are therefore denoted as {h𝐺𝐶, h1, . . . , h𝑚}. + +Geolocation l +GCL Loss +MGM Loss +Single-Modal MLM Loss +Multi-Modal MLM Loss +Multi-Modal MGM Loss +(106.458,29.563) +(Geo Input) +(Geo Input) +(Text Input) +(Text & Geo Input) +(Text & Geo Input) +MASK +el +hcLs +h +y +htm +hsEP +ei +Add & Norm +MASK +hGC +01 +Oi +On +eGC +Geographic Objects +en +Feed-Forward +e1 +h1 +Multi-Modal Interaction +× Multi-Layer +Add & Norm +et +hi +e? +ei +Multi-Head Attention +[pq +ei +CLS +t1 +tj +tm +SEP +h1 +ni +el +hn +nn +Relation: COVERED +en +Word Embeddings +Geographic Encoder +OSMID: 3119 +Geographic +Shape: POLYGON +Encoder +Text t +Geolocation lA Multi-Modal Geographic Pre-Training Method +Preprint, +4.1.2 +Training. We design two tasks to train geographic encoder +and it is fixed in later uses, i.e., masked geographic modeling (MGM) +and geographic contrastive learning (GCL). +MGM. Like the widely use masked language modeling (MLM) [5], +MGM aims at predicting masked geographic features, i.e., OSM IDs, +geometric types, each side of the substituted rectangle, relation +types, and relative positions. The MGM loss 𝐿𝑀𝐺𝑀 is calculated by +summing up the masked loss of all features. +GCL. This task is related to multiple geolocations {𝑙𝑝𝑞 +1 , . . . ,𝑙𝑝𝑞 +𝑏𝑠 } +in a batch of size 𝑏𝑠. We begin with the definition of the real-world +geographic distance matrix H ∈ R𝑏𝑠×𝑏𝑠 defined as: +H𝑖,𝑗 = 𝜎 �−∥ℎ𝑎𝑣𝑒𝑟𝑠𝑖𝑛𝑒(𝑙𝑝𝑞 +𝑖 ,𝑙𝑝𝑞 +𝑗 )∥N +�,𝑖, 𝑗 ∈ {1, . . . ,𝑏𝑠},𝑖 ≠ 𝑗, +(10) +whereℎ𝑎𝑣𝑒𝑟𝑠𝑖𝑛𝑒 is the haversine function [23] that calculates spher- +ical distance between geolocations, ∥ · ∥N is gaussian normalization +function, and 𝜎 is sigmoid function that maps distance to range +(0, 1). As the latent distance between embeddings in the output +space should correspond to their real-world geographic distance, +we use ℎ𝐺𝐶 as the representation of geolocation 𝑙𝑝𝑞 with GC and +calculate the latent distance matrix ˜H ∈ R𝑏𝑠×𝑏𝑠 as: +˜H𝑖,𝑗 = ⟨∥h𝑖 +𝐺𝐶 ∥𝐿2, ∥h𝑗 +𝐺𝐶 ∥𝐿2⟩ +(11) +where ⟨·⟩ denotes the doc-product function and ∥ · ∥𝐿2 is 𝐿2 normal- +ization function. We use KL-divergence to measure the similarity +between H and ˜H. GCL loss 𝐿𝐺𝐶𝐿 is then calculated by: +𝐿𝐺𝐶𝐿 = +𝑏𝑠 +∑︁ +𝑖=1 +𝐷𝐾𝐿 +�𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(H𝑖) ∥ 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥( ˜H𝑖)� +(12) +where 𝐷𝐾𝐿(· ∥ ·) denotes the KL-divergence, and the 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥 +function is applied to transform H𝑖 and ˜H𝑖 to a distribution. +The training loss 𝐿𝑔 of geographic encoder is thus calculated by: +𝐿𝑔 = 𝐿𝑀𝐺𝑀 + 𝐿𝐺𝐶𝐿 +(13) +Using such an training process, geographic encoder is capable of +modeling GC in a given GIS. +4.2 +Multi-Modal Pre-Training +The input of MGeo pre-training is a pair of text and geolocation (𝑡, +𝑙). The pre-training data can come from diverse sources, e.g., click +of users or position of delivery clerks. The multi-modal training +aims at aligning these two modals into one latent space. Word +embeddings are used to map text into a sequence of vectors. The +geographic encoder provides the GC embeddings given 𝑙. The two +embeddings are then concatenated together and fed into multi-layer +bidirectional Transformers. +We use three tasks to learn interaction between GC and text, +i.e., single-modal MLM, multi-modal MLM, and multi-modal MGM. +These tasks are trained in turns. Single-modal MLM is the original +MLM task used in BERT, which randomly masks and replaces the +input text with 𝑀𝐴𝑆𝐾 token. The outputs of geographic encoder are +removed for single-modal MLM. While multi-modal MLM predicts +the masked token relying on the entire GC and part of textual +information. Multi-modal MGM randomly masks and replaces the +input geographic features with 𝑀𝐴𝑆𝐾 and predicts them relying +on entire textual information and part of GC. +(A) Bi-Encoder. +(B) Cross-Encoder. +Figure 3: MGeo can use both (A) bi-encoder and (B) cross- +encoder architectures to measure relevance between query +and POI. Dashed line indicates that geolocation of query is +optional. ⊕ denotes element-wise addition. +4.3 +Relevance Measurement +MGeo can use both bi-encoder and cross-encoder architectures, as +shown in Figure 3. Bi-encoder encodes query and POI separately +for efficiency issues. It can be used in both retrieval and ranking +phases. In practice, the GC of a POI or query is encoded by geo- +graphic encoder. Since user location is not always available due +to privacy issues or limited hardware, the GC of query can be ab- +sent. The outputs are then concatenated with word embeddings. +Transformer-based multi-modal interaction module then produces +hidden states as final representations. We compute the similarity +score of a query and POI pair by the cosine similarity between +their 𝐶𝐿𝑆 representations, i.e., ˆh𝑝 and ˆh𝑝. Bi-encoder calculates +similarity scores between a query and all the POIs for retrieval task. +Different from bi-encoder, cross-encoder concatenates every +query-POI pair together before being fed to multi-modal interaction +module. Cross-encoder allows fine-grained token-level interaction +between query and POI, it usually provides a more accurate esti- +mation of relevance but is less efficient. Therefore, cross-encoder +is only used in ranking phase as usual. The GC of query or POI +is encoded separately by geographic encoder. The GC of query +is also optional. We concatenate query textual embeddings, POI +textual embeddings, query GC embeddings (optional), and POI GC +embeddings together, which are then fed to multi-modal interaction +module. Particularly, we use geographic discriminator to facilitate +geographic comparison between GC of query and POI. Geographic +discriminator adds embeddings to outputs of geographic encoder to +distinguish query GC from POI GC. Like the segment embeddings + +Similarity Score +CLS +MGeo +MGeo +Multi-Modal Interaction +Multi-Modal Interaction +Word Embeddings +Geographic Encoder +Word Embeddings +Geographic Encoder +Query Text tq +Query Geolocation 1q +POI Text tp +POIGeolocation[p +OptionalSimilarity Score +MGeo +Multi-Modal Interaction +Word Embeddings +Geographic Encoder +Geographic +Geographic +Segment 1 +Segment 2 +Discriminator1 +Discriminator 2 +Query Text tq +POI Text tp +Query Geolocation [q +POI Geolocation [p +OptionalPreprint, +Ding et al. +Table 2: Statistics of different query types. +Query Type +# Query +Address +81,286 +Street No. +6,013 +Colloquial +2,701 +Total +90,000 +Table 3: Statistics of train/dev/test splits. +# Query +# Candidate POI +Train +50,000 +20 +Dev +20,000 +40 +Test (Ranking) +20,000 +40 +Test (Retrieval) +2,849,754 +Table 4: Statistics of geographic objects. +Line +Polygon +Covered +Nearby +Covered +Nearby +Query +0.005 +4.4 +0.7 +14.2 +POI +0.003 +3.7 +0.6 +10.4 +in BERT, embeddings of geographic discriminator are randomly +initialized and trainable. We fed the hidden states of 𝐶𝐿𝑆 ˆh𝑝𝑞 +𝐶𝐿𝑆 to a +multi-layer perceptron (MLP) to produce similarity scores. +5 +THE GEOTES BENCHMARK +In this section, we introduce our proposed large-scale benchmark +GeoTES, which stands for Geographic TExtual Similarity. It is the +first open-source benchmark for query-POI matching. The POIs are +obtained from the open-source OSM and the queries are manually +generated by annotators to prevent privacy issues. +5.1 +Annotation Process +In this version of GeoTES, all the POIs are located in Hangzhou. 20 +annotators and 4 experienced experts are asked to annotate three +types of queries based on the POIs, as described in Section 3. Ta- +ble 2 gives the statistics of these query types, which follows the +distribution of our online LBS. In OSM, each POI comes with a +geographic location under the WGS84 coordinate system.3 Neigh- +bouring POIs of the OSM POIs from several open-accessed map +services are selected by the annotators to enrich the diversity of +POI description and also serve as hard negatives. To simulate the +queries’ location in real scenes, the annotators are asked to ran- +domly select a location within 1km of corresponding POI for 50% +of the queries and randomly select a location in the city for the rest +queries. All the annotators have adequate linguistic knowledge and +educational/cultural background to produce appropriate queries. To +3https://wiki.openstreetmap.org/wiki/Converting_to_WGS84 +Table 5: Model sizes of pre-trained and fine-tuned models. +Pre-Train +Fine-Tune +BERT-DA +118M +102M +BERT-MGeo +213M +129M +eliminate biases during the annotation process, they are instructed +with detailed annotation principles. One quality inspector ensures +that each of the queries has one matched POI. +5.2 +Benchmark Statistics +GeoTES has a total number of 90,000 queries with an average length +of 17.2 and 2,849,754 POIs with an average length of 13.7. We extract +the geographic surrounding objects for the queries and POIs from +OSM. There are 21,950 lines and 65,722 polygons in our extracted +geographic objects. As given in table 4, each query and POI has +more relations to polygons than lines. The benchmark is randomly +split in to train, development, and test sets, as shown in Table 3. For +the train, development, and ranking test sets, we provide a list of +candidate POIs and ensure that one exact matched POI is contained. +The retrieval test set use the same queries as the ranking test set +while no candidate POI should be provided. Therefore, we believe +that the GeoTES presents a reliable and challenging dataset for +benchmarking retrieval and ranking models. +6 +EXPERIMENT +In this section, we compare the proposed MGeo with several strong +baselines on GeoTES. +Task. The experiments are conducted on two tasks, i.e., ranking +and retrieval. The two tasks use the same train, development, and +test sets as shown in Table 3. A list of candidate POIs that contains +the relevant one is provided for the ranking task. Both bi-encoder +and cross-encoder are evaluated on ranking task. Since retrieval +task requires searching the full POI corpus, and cross-encoder needs +too much computing resources to complete retrieval task, only bi- +encoder is evaluated on retrieval task. +Evaluation metric. Following previous IR work [24], we use Re- +call and Mean Reciprocal Rank (MRR) at top 𝑘 ranks to evaluate +the performance on both tasks. Recall@𝑘 calculates the proportion +of queries that have the relevant POI contained in the top-𝑘 candi- +dates, and MRR@𝑘 calculates the averaged reciprocal of the rank at +which the relevant POI is placed. We report the evaluation scores +on the test set of models that perform best on the development set +during training. +PTM Baseline. We first evaluate the performance of four widely +used PTMs with the base model size on GeoTES, including BERT [5], +RoBERTa4 [21], ERNIE 3.0 [31], and StructBERT [35]. We further +apply domain-adaptive pre-training techniques (DA) on BERT and +another top-performing model. DA is a widely used single-modal +pre-training baseline [9]. For a fair comparison, domain corpus +used in DA is the same as that used in our proposed multi-modal +4https://huggingface.co/clue/roberta_chinese_base + +A Multi-Modal Geographic Pre-Training Method +Preprint, +Table 6: Ranking results of bi-encoder and cross-encoder. Bold indicates the best of each column. +Bi-Encoder +Cross-Encoder +PTM +Recall@1 +Recall@3 +Recall@5 +MRR@5 +Recall@1 +Recall@3 +Recall@5 +MRR@5 +BERT +58.83 +79.40 +86.24 +69.60 +81.52 +91.11 +94.10 +86.53 +RoBERTa +68.52 +85.41 +90.25 +76.15 +83.20 +93.09 +95.77 +88.30 +ERNIE +50.24 +72.97 +81.87 +62.40 +79.45 +90.04 +93.40 +85.02 +StructBERT +69.09 +86.29 +91.09 +77.96 +83.51 +93.21 +95.67 +88.53 +BERT +DA +72.49 +89.18 +93.48 +81.03 +83.24 +92.92 +95.63 +88.25 +StructBERT +74.30 +89.78 +94.06 +82.28 +83.65 +93.33 +95.92 +88.61 +BERT +MGeo +w/o query GC +74.86 +90.61 +94.53 +82.93 +85.11 +94.42 +96.75 +89.86 +StructBERT +75.37 +89.99 +93.96 +82.89 +84.72 +93.85 +96.16 +89.40 +BERT +MGeo +76.04 +91.24 +95.18 +83.85 +85.89 +95.48 +97.48 +90.74 +StructBERT +76.07 +90.68 +94.50 +83.57 +86.49 +95.55 +97.62 +91.10 +Table 7: More ranking results of bi-encoder and cross- +encoder baselines. +Recall@1 +Bi-Encoder +DSSM [39] +34.59 +DPAM [39] +44.15 +PALM [39] +45.51 +BERT +58.83 +ColBERT [15] +62.36 +Poly-Encoder [13] +49.87 +BERT-MGeo +76.04 +Cross-Encoder +BERT +81.52 +ERNIE-GeoL [11] +82.94 +BERT-MGeo +85.89 +geographic pre-training (MGeo), except that MGeo has additional +GC along with query and POI. +6.1 +Hyperparameter +The architecture of the multi-modal interaction module is multi- +layer transformers. The model sizes are listed in Table 5. +6.1.1 +Geographic Encoder. All geographic feature embeddings are +set to 256. The discretization factor 𝑘 is 10 and the grid number +𝑁 is 2000. Geographic encoder has 4 layers of transformer with +256 hidden sizes. The mask probability is 0.15. The training batch +size is 512. We use AdamW as optimizer with learning rate being +1e-4, weight decay being 0.02. We train geographic encoder for 30 +epochs and take the last epoch checkpoint. +6.1.2 +Pre-Training. The training batch size is 512. We use AdamW +as optimizer with learning rate being 5e-5, weight decay being 0.02. +We train for 10 epochs and take the last epoch checkpoint. +6.1.3 +Downstream Task. For bi-encoder models, every training step +has 56 queries, each has 20 candidates. We use AdamW as optimizer +with learning rate being 5e-5, weight decay being 0.02. Specifically, +Table 8: Retrieval results of bi-encoder. +BERT +BERT-DA +BERT-MGeo +Recall@1 +21.70 +51.76 +52.70 +Recall@3 +27.06 +58.36 +60.28 +Recall@5 +29.32 +60.82 +63.39 +Recall@20 +35.70 +67.08 +70.49 +Recall@50 +40.30 +71.61 +75.00 +Recall@100 +44.02 +74.74 +78.29 +MRR@5 +24.58 +55.29 +56.79 +MRR@10 +24.98 +55.71 +57.25 +ERNIE and StructBERT don’t converge in this learning rate, we +change it to 5e-6. We train geographic encoder for 10 epochs. +For cross-encoder models, every training step has 24 queries and +the learning rate for RoBERTa is 5e-6. Other settings are the same +as bi-encoder. +6.2 +Ranking +Table 6 gives the ranking results of both bi-encoder and cross- +encoder. As the original StructBERT outperforms the other generic +PTMs, it is used for further DA. The generic PTMs directly fine- +tuned on the downstream tasks show a low performance, which +indicates that these two tasks are challenging. Since cross-encoder +can make fine-grained interactions among input features, while +bi-encoder only interacts with the 𝐶𝐿𝑆 representations for the sake +of efficiency, cross-encoder generally outperforms bi-encoder by a +large margin. +By applying DA on bi-encoder, PTMs could gain an advantage +over the generic ones. However, DA models consider only the tex- +tual modality and neglect the other geographic modal. Through +multi-modal pre-training, MGeo without query GC raises 2.37% +(resp., 1.07%) point of Recall@1 on BERT-DA (resp., StructBERT-DA) +by bridging the gap between query text and POI GC. After being +accompanied by query GC, MGeo further shows a 3.55% (resp., +1.77%) improvement in Recall@1 over DA models with the help of +incorporating correlations between query GC and POI text, as well + +Preprint, +Ding et al. +Table 9: Inference time (second) of bi-encoder and cross- +encoder models. +Bi-Encoder +Cross-Encoder +BERT-DA +0.0219 +0.0396 +BERT-MGeo w/o query GC +0.0205 +0.0414 +BERT-MGeo +0.0269 +0.0466 +as between query GC and POI GC. It is worth noting that GC of half +the training and test queries are noises to simulate the arbitrary +geolocation of users, as described in Section 5. The results show +MGeo’s capability of denoising and it may gain more improvements +if the queries have more precise geolocations. +In cross-encoder, MGeo also shows superiority over baselines. +DA brings fewer benefits on PTMs than it does in bi-encoder, i.e., +1.72% on BERT and 0.14% on StructBERT. However, improvements +brought by incorporating the new geographic modal are consistent. +MGeo without query GC gains 1.87% (resp., 1.07%) Recall@1 on +BERT-DA (resp., StructBERT-DA). Together with query GC, MGeo +boost DA models by 2.65% (resp., 2.84%) in Recall@1, showing +effectiveness of multi-modal interaction. +6.2.1 +More Baseline Comparison. Besides the PTM baselines, we +also add more query-POI matching baselines, including two SOTA +text-matching models, i.e., ColBERT [15] and Poly-Encoder [13]. +ColBERT uses a late interaction architecture to enhance bi-encoder +model. Similarly, Poly-Encoder uses attention mechanism to capture +richer interactions between query and POI. Detailed introductions +of DSSM, DPAM, and PALM can be found in [39]. ERNIE-GeoL is +a strong PTM cross-encoder baseline introduced in [11]. Since the +data and code of ERNIE-GeoL are not released, we only adopt the +pre-training objectives. The results on the ranking task are shown +in Table 7. For bi-encoder, BERT-MGeo still outperforms ColBERT +and Poly-Encoder, which capture more fine-grained interactions +between query and POI. For cross-encoder, ERNIE-GeoL uses spe- +cific pre-training objectives to capture static geographic knowledge +and outperforms BERT. While BERT-MGeo capture dynamic GC +and outperforms ERNIE-GeoL. +6.3 +Retrieval +Bi-encoder is also evaluated on retrieval task. Since retrieval task +focuses on finding the relevant POIs rather than ranking the correct +POI at the top, table 8 reports Recall and MRR metrics. +Compared to BERT-DA, MGeo improves 3.41% Recall@20. The +results demonstrate that the effectiveness of MGeo in bi-encoder +architecture stays consistent when the size of candidates becomes +100,000 times larger. +6.4 +Inference Time +The inference time on 1 NVIDIA V100 GPU of bi-encoder and cross- +encoder models is listed in Table 9. For bi-encoder, we only count +the time of query encoding, since the document can be encoded in +advance in many industrial scenarios. We use 26 queries and 1040 +documents for inference. +Table 10: Influence of different geographic object types. +Bi-Encoder +Cross-Encoder +Recall@1 +MRR@5 +Recall@1 +MRR@5 +Line +74.56 +82.57 +83.71 +88.88 +Polygon +74.26 +82.51 +84.84 +89.85 +Both +76.04 +83.85 +85.89 +90.74 +Figure 4: Ranking MRR@5 for different percentage of query +with GC. +6.5 +Ablation Study +Since we use the same bi-encoder models for both retrieval and +ranking tasks, the ablation study is mainly conducted on ranking +task of BERT-based models. +6.5.1 +Geographic Object. We first study the influence of training +queries with GC. We randomly remove GC of the same proportion +from the training, development, and test queries. As shown in +Figure 4, the performance is impaired when a small proportion of +queries contain GC. This decrease comes from a larger proportion +of noise. Taking 30% of queries having GC as example, there are +already 15% GC are noises (half GC are randomly selected). Since +it is difficult to distinguish query without GC from query without +geographic object (but with geolocation), the rest queries without +GC can be considered as noises too. Thus we have in total 75% +queries with noisy GC, which damages model performance. When +noises proportion becomes smaller than 65% (70% query with GC), +the performance is better than training without query GC. +The influence of different geographic object types is reported in +Table 10. There is not a huge gap between line and polygon for bi- +encoder, while cross-encoder can perform better with only polygon +than only line, as there are more polygons than lines in the GIS. +This also suggests that cross-encoder is better at capturing the fine- +grained correlations than bi-encoder. Nevertheless, using either +line or polygon is better than the single-modal baselines. Besides, + +Bi-Encoder +Cross-EncoderA Multi-Modal Geographic Pre-Training Method +Preprint, +(A) Bi-Encoder. +(B) Cross-Encoder. +Figure 5: Ranking Recall@1 for (A) bi-encoder and (B) cross- +encoder with different query types. +bi-encoder and cross-encoder can have a better performance when +the two types of geographic objects both present. +6.5.2 +Query Type. Figure 5 shows the performance on three query +types, i.e., address, street number, and colloquial. Bi-encoder models +perform best on address description, while cross-encoder models +perform best on street number description. This suggests that cross- +encoder is better at capturing fine-grained correlations. Colloquial +query contains many daily expressions, which rarely appear in +domain corpus. Thus BERT-DA is even worse than BERT on it. +However, the use of GC help reduce this shortcoming of DA. +6.5.3 +Amount of Training Data. We study the performance of MGeo +with different amounts of training data. As shown in Figure 6, the +dashed line is used for representing BERT-DA and the dotted line +for the original BERT. With only 30% of training data, the bi-encoder +and cross-encoder using MGeo can outperform the BERT baseline +by a large margin. +6.5.4 +Query Incompleteness. POI suggestion also plays an impor- +tant role in LBS, where the name of POIs are listed when the input +(A) Bi-Encoder. +(B) Cross-Encoder. +Figure 6: Ranking Recall@1 for (A) bi-encoder and (B) cross- +encoder with different amounts of training data. +(A) Bi-Encoder. +(B) Cross-Encoder. +Figure 7: Ranking Recall@1 for (A) bi-encoder and (B) cross- +encoder with different percentage of query incompleteness. +is unfinished. To simulate such scenario, we also evaluate MGeo +on incomplete queries by truncating the trailing characters. +Figure 7 shows the performance with different truncation ratio +of the test queries. The results demonstrate that bi-encoder using +MGeo could outperform the BERT baseline with full queries with a +small truncation ratio. Whereas the cross-encoder could not, since +the semantic similarity is more important for cross-encoder. +7 +CONCLUSION +In this paper, we formalize an important concept GC, which is +indispensable for real-world human POI exploration process. We +propose a multi-modal geographic language model MGeo, which +considers GC as a new modal. Therefore, GC can be represented to- +gether with text. In addition, we build a new open-source large-scale +benchmark GeoTES to facilitate further research on the query-POI +matching topic. Extensive experiments are conducted to evaluate +our proposed method on the state-of-the-art PTMs, and the detailed +analyses demonstrate that MGeo can significantly outperform other +baselines. Even though geolocation of user may be absent and query +has no GC, MGeo can still obtain improvements over the baselines, +showing its capability of modeling text-to-text, GC-to-GC and text- +to-GC correlations. For future work, other modalities like POI image +can be further explored, as well as more inventive geographic en- +coder. Besides, our proposed GC modeling has the potential to boost +all geography-related tasks. + +BERT-MGeo +BERT-DA +BERTBERT-MGeo +BERT-DA +BERTAddress +Street No. +ColloquialAddress +Street No. +ColloquialBERT-MGeo +BERT-DA +BERTBERT-MGeo +BERT-DA +BERTPreprint, +Ding et al. +REFERENCES +[1] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexan- +der Kirillov, and Sergey Zagoruyko. 2020. End-to-End Object Detection with +Transformers. In Computer Vision – ECCV 2020. +[2] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed El Kholy, Faisal Ahmed, Zhe Gan, +Yu Cheng, and Jingjing Liu. 2020. UNITER: UNiversal Image-TExt Representation +Learning. 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In Proceedings of the +AAAI Conference on Artificial Intelligence. + diff --git a/RNE3T4oBgHgl3EQfDAny/content/tmp_files/load_file.txt b/RNE3T4oBgHgl3EQfDAny/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8e4a8478ebcf1e3ac4d795475fc5e14f05002f4 --- /dev/null +++ b/RNE3T4oBgHgl3EQfDAny/content/tmp_files/load_file.txt @@ -0,0 +1,888 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf,len=887 +page_content='A Multi-Modal Geographic Pre-Training Method Ruixue Ding†∗, Boli Chen†∗, Pengjun Xie†, Fei Huang†, Xin Li‡, Qiang Zhang‡, Yao Xu‡ †DAMO Academy, Alibaba Group ‡Gaode Map, Alibaba Group {ada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='drx,boli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='cbl,chengchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='xpj,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='huang,beilai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='bl,muxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='zq,xuenuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='xy}@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='com ABSTRACT As a core task in location-based services (LBS) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', navigation maps), query and point of interest (POI) matching connects users’ in- tent with real-world geographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Recently, pre-trained models (PTMs) have made advancements in many natural lan- guage processing (NLP) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Generic text-based PTMs do not have enough geographic knowledge for query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To overcome this limitation, related literature attempts to employ domain-adaptive pre-training based on geo-related corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' How- ever, a query generally contains mentions of multiple geographic objects, such as nearby roads and regions of interest (ROIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The geographic context (GC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', these diverse geographic objects and their relationships, is therefore pivotal to retrieving the most rele- vant POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Single-modal PTMs can barely make use of the important GC and therefore have limited performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In this work, we pro- pose a novel query-POI matching method Multi-modal Geographic language model (MGeo), which comprises a geographic encoder and a multi-modal interaction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGeo represents GC as a new modality and is able to fully extract multi-modal correlations for accurate query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Besides, there is no publicly avail- able benchmark for this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In order to facilitate further research, we build a new open-source large-scale benchmark Geographic TExtual Similarity (GeoTES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The POIs come from an open-source geographic information system (GIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The queries are manually generated by annotators to prevent privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Compared with several strong baselines, the extensive experiment results and de- tailed ablation analyses on GeoTES demonstrate that our proposed multi-modal pre-training method can significantly improve the query-POI matching capability of generic PTMs, even when the queries’ GC is not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Our code and dataset are publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='com/PhantomGrapes/MGeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Language models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Similarity mea- sures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Business intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' KEYWORDS query-POI matching, multi-modal, language model, geographic context, benchmark ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Preprint, © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='nnnnnnn Figure 1: A typical query-POI matching procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Reference Format: Ruixue Ding, Boli Chen, Pengjun Xie, Fei Huang, Xin Li, Qiang Zhang, Yao Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' A Multi-Modal Geographic Pre-Training Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 1 INTRODUCTION As an essential function of location-based services (LBS) like naviga- tion maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', Google Maps), ride-hailing applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', Uber), and food delivery platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', Uber Eats), query and point of interest (POI) matching aims to find a list of candidate POIs based on users’ specific or implicit intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since the candidate results are crucial for providing real-world geographic information to the users, which directly impacts the navigation, routing, and ordering process, effective and accurate query-POI matching is indispensable for delivering a satisfactory user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' A typical query-POI matching procedure is illustrated in Figure 1, which consists of a two-stage retrieve-then-rank pipeline [37, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In specific, given a query, the lightweight retriever first produces an initial set of can- didate POIs by searching a massive database, then the ranker sorts the most relevant candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' This kind of architecture is widely adopted in information retrieval (IR) systems on account of the efficiency-effectiveness trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Recent literature on natural language processing (NLP) as well as IR shows a flourishing advancement of pre-trained models (PTMs), arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='04283v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='CL] 11 Jan 2023 重庆南开 日 中学 Chongqing Nankai Secondary School 重庆市名 校联合中 学 Chongqing Unite Secondary Scho 三峡广场 Sanxia Square School gate on underground road Retrieval RankingPreprint, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' notably in semantic textual similarity (STS) and open-domain ques- tion answering (QA) [3, 14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since continued self-supervised training on domain-specific corpus is shown to be effective for PTMs [9], various domain-adaptation methods have lately been proposed to inject the geographic knowledge based on geo-related textual data and relevant user behavioral data [10, 11, 20, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Al- though better at capturing the semantic similarity than the generic PTMs, these methods can barely make use of the more important circumstantial geographic context (GC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', the diverse geographic objects and their relationships from the geographic information system (GIS) detailed in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Specifically, the geographic objects consist of roads represented as lines and regions of inter- est (ROIs) represented as polygons, the relationship includes near, covered, and their relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As the query usually mentions multiple geographic objects in the background, extracting the correlations between these objects is necessary for accurate query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For example, given the query "school gate on underground road", as shown in Figure 1, several relevant POIs are retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The nearest "underground road" to the user is the "Nankai Underground Rd", and the "Nankai Sec- ondary School" has a gate (c) on the "Underground Rd".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Therefore, the most matched POI should be the gate (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The problem is that the "Nankai Secondary School" is formally located on the "Shapingba S St" with its main gate (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Its side gate (c) is not recorded in the GIS as located on the "Underground Rd".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' It should also be noticed that the user is currently in the "Sanxia Square", which has a gate (b) located on the "Underground Rd".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The semantic textual similarity alone is not enough to distinguish these two hard negatives (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Moreover, the gate (d) of the "United Secondary School" is the closest school gate to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Simply considering the relative position of the user and the POI will match the wrong gate (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Only by taking the entire GC into consideration can we find the correct gate (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To this end, we propose a novel method Multi-modal Geographic language model (MGeo), which bridges the modal gap between se- mantics and GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGeo consists mainly of a geographic encoder and a multi-modal interaction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The geographic encoder makes use of the GC by representing it as a new modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The multi-modal interaction module then incorporates the geographic features with the semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGeo makes use of the textual, geo- graphic, and cross-modal interactions between queries and POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since the interaction module is compatible with queries that have no GC, it is optional to provide the queries’ geolocation as many applications may require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As a result, rich correlations among tex- tual and geographic modalities can be fully extracted to ensure the quality of query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In addition, there exists no public unencrypted benchmark for query-POI matching mostly due to privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Large publicly available corpus could lead to many breakthroughs in research, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', MS MARCO [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Intending to facilitate further research on this topic, develop robust techniques, and track progress, we introduce Geographic TExtual Similarity (GeoTES), which is an open-source large-scale benchmark for query-POI matching with GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The POIs come from the open-source GIS OpenStreetMap (OSM)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To prevent 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='org privacy issues, the queries are manually generated by annotators thus do not require encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Our major contributions are highlighted as follows: We formalize the important concept GC for the query-POI matching problem and propose a novel method MGeo that uses geographic encoder to represent it as a new modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use multi-modal interaction module to incorporate the correlations among textual and geographic modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Be- sides, MGeo is compatible with queries that have no GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We develop a new open-source large-scale benchmark named GeoTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The POIs come from an open-source GIS and the queries are manually generated by annotators to prevent privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Compared with strong baselines, the experiment results demonstrate that our proposed MGeo can significantly im- prove the query-POI matching capability of generic PTMs, even when no GC is provided for the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Relevance Model Traditional approaches for retrieving documents from large corpus generally use exact term-level matching, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', Okapi Best Match- ing (BM25) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Despite such heuristic retrievers have low latency via inverted list data structure, their measurement of similarity is only based on document statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' On the other hand, the latent models can correlate semantically similar terms and reduce the matching dimensionality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', Partial Least Square (PLS) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Lat- terly, Deep neural network (DNN) models have been introduced to IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For example, Deep Structured Semantic Model (DSSM) [12] measures the relevance of queries and documents in a semantic vector space by computing their cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Along with the success of PTMs in NLP, studies on IR have also made remark- able progress [7, 14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' On account of the efficiency-effectiveness trade-off, there are two major architectures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', bi-encoder and cross-encoder [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-encoder allows efficient indexing [4, 26] and is usually used in the retrieval system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In contrast, cross-encoder concatenates the query and document to perform cross-interaction over all input terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Although cross-encoder can provide a more ac- curate estimation of relevance, it needs more computing resources and is usually used only in the ranking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGeo can use either the bi-encoder or cross-encoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Multi-Modal Representation Learning Following the tremendous success of various pre-training tech- niques in NLP, a lot of Transformer-based models are proposed for other modalities, such as compute vision (CV) [1, 6, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Except for single-modal, recent studies also show the derivative models have great potential in multi-modal representation learning [2, 16, 18, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For example, CLIP [25] converts classification to a retrieval task and enables zero-shot learning via large-scale multi-modal pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In addition to image, layout of document and table can also be represented as different modalities [33, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In this paper, we pro- pose a novel multi-modal geographic pre-training method, which represents the GC as a new modality for query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' A Multi-Modal Geographic Pre-Training Method Preprint, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Query-POI Matching Previous work focuses on modeling the relative position between queries and POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Based on DSSM, PALM [39] obtains the positional relationship of queries and POIs from coordinate-based and kernel- based location embeddings, and incorporates the relationship with semantic similarity for POI retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' STDGAT [38] further takes multiple spatiotemporal factors into consideration via dual graph at- tention network when quantifying the query-POI relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' On ac- count of the ubiquity of PTMs in NLP, domain-adaptive pre-training methods have been proposed to inject extralinguistic knowledge into the generic PTMs [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Typically, GeoL [11] makes use of the static geographic knowledge based on user behavioral (search logs), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', geocoding [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Despite the domain-adapted PTMs may be better at capturing the semantic similarity than the generic PTMs, they are still limited by ignoring the GC in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In addition, since the public benchmark can facilitate further research and play an important role in the development of robust techniques, we also establish a reliable large-scale query-POI match- ing benchmark GeoTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 3 PRELIMINARY We first introduce the formal description of the query-POI matching problem, as well as some important definitions related to GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Table 1 gives the frequently used notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Let 𝑃 be the set of POIs𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑃 can either contain dozens of candidate POIs or a large number of POIs in the massive database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Each POI 𝑝 consists of a textual description 𝑡𝑝 and its geolocation 𝑙𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The textual description of the POI 𝑡𝑝 contains its formal address and name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Let 𝑞 denote a query made by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The textual description of the query 𝑡𝑞 belongs to three types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', common address description, formal street number description, and casual colloquial description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The street number query contains standard numerical designation for a target POI, while the address query does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The colloquial query uses spoken language and may contain colloquial noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The query’s geolocation 𝑙𝑞 can be the users’ geolocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' When the user searches for another area using the map, 𝑙𝑞 is the center location displayed on screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Furthermore, 𝑙𝑞 may or may not be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We denote geolocation of a POI or query as 𝑙𝑝𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Query-POI matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Given the POI set 𝑃 and a user’s query 𝑞 in LBS, we aim to estimate the POI 𝑝 ∈ 𝑃 that best matches the user’s intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We define two tasks based on the size of 𝑃, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', ranking and retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Specifically, for the ranking task, 𝑃 is a list of candidate POIs, where the best-matched one is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As for the retrieval task, 𝑃 is the massive database that contains all POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since cross- encoder is inefficient for large size of 𝑃, it only runs on the ranking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-encoder can run on both the ranking and retrieval tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geographic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' GIS is constructed on spatial data that defines the real-world geometric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Let G be the spatial database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Each geographic object 𝑜 ∈ G with 𝑚 vertices is described as a sequence of geolocation {𝑙𝑜 1,𝑙𝑜 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑙𝑜𝑚} and characterized by its shape 𝑜𝑠 ∈ {𝐿𝐼𝑁𝐸, 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Specifically, 𝐿𝐼𝑁𝐸 represents the real-world road and 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 represents the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Table 1: Table of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Notation Description 𝑃, 𝑝 The POI set and the POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑞 The query of user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑜 The geographic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑜𝑠 The shape of geographic object, ∈ {𝐿𝐼𝑁𝐸, 𝑃𝑂𝐿𝑌𝐺𝑂𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑜𝑚 The position of 𝑜 in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑡 The textual description of POI or query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑙 = (𝑙𝑛𝑔,𝑙𝑎𝑡) The geolocation represented by longitude and latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑙𝑝𝑞 The geolocation of a POI or query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑙𝑜 A vertex of 𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ˜𝑜 The rectangle that approximates the shape of 𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑟𝑡 The relation type ∈ {𝑁𝐸𝐴𝑅,𝐶𝑂𝑉 𝐸𝑅𝐸𝐷 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑟𝑝 The relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Here we use 𝑚 to denote the number of vertices in 𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Note that given the geolocation of the POI or the query, we can form a list of nearby geographic objects {𝑜1,𝑜2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑜𝑛} sorted by distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 𝑜1 is the nearest geographic object to the POI or query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑛 is used to denote the number of geographic objects for a geolocation 𝑙𝑝𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We export OSM to PostGIS2 and get the Geographic Context (GC) of a geolocation from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geographic Context (GC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Given the geolocation 𝑙𝑝𝑞 of a POI or query, where 𝑙𝑝𝑞 is represented by a geographic coor- dinate (𝑙𝑛𝑔,𝑙𝑎𝑡), the GC is characterized by the relationships between 𝑙𝑝𝑞 and its 𝑛 geographic objects {𝑜1,𝑜2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑜𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Formally, the rela- tion type 𝑟𝑡 ∈ {𝑁𝐸𝐴𝑅,𝐶𝑂𝑉𝐸𝑅𝐸𝐷} indicates whether 𝑙𝑝𝑞 is inside 𝑜𝑖 or at a distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The relative position 𝑟𝑝 depicts a more detailed positional relationship between 𝑙𝑝𝑞 and 𝑜𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' When searching for a target POI, a user usually explores nearby circumstantial spatial data and mentions multiple related geographic objects in the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Besides, the intrinsic characteristics of geo- graphic objects are also important GC features (described in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Therefore, the GC is pivotal to ensuring the quality of query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 4 METHOD In this section, we present the detailed architecture and pre-training process of MGeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Following state-of-the-art multi-modal meth- ods [2, 16, 19], MGeo is composed of a geographic encoder and a multi-modal interaction module, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The full training process of MGeo consists of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' First, we train ge- ographic encoder alone to learn representations of GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The trained geographic encoder is fixed in the following stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Then, text- geolocation pairs are used to pre-train MGeo in a multi-modal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' By modeling geographic objects along with text and pre-training with massive text-geolocation pairs, MGeo successfully aligns these two modals into a same latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Lastly, MGeo is fine-tuned on ranking and retrieval tasks and gains significant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Geographic Encoder The geolocation alone is meaningless unless it has GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Taking a geolocation 𝑙 as input, geographic encoder maps the GC as a new modality to dense representations, which contains features of the surrounding geographic objects {𝑜1,𝑜2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑜𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 2https://postgis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='net/ Preprint, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 2: Architecture of MGeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Left part shows encoding and pre-training process of geographic encoder and right part shows the multi-modal pre-training process of MGeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Word embeddings of text t and GC representations of geographic encoder h are concatenated together and fed to multi-modal interaction module, which produces final representations ˆh𝑡 for each text token and ˆh𝑙 for each geographic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geographic encoder can extract the relationships between geolocation and geographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For a geographic ob- ject 𝑜𝑖, a one-hot function is used to encode the categorical relation type 𝑟𝑡 𝑖 as a numeric array and to obtain its corresponding embed- dings e𝑡 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To simplify the relative position 𝑟𝑝 𝑖 , we form a rectangle ˜𝑜𝑖 of similar size to approximate the shape of 𝑜𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Each side of the substituted rectangle (left, bottom, right, and top) is defined as: ˜𝑜𝑙𝑒𝑓 𝑡 𝑖 = 𝑚𝑖𝑛�{𝑙𝑛𝑔𝑜𝑖 𝑗 }𝑗 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=',𝑚𝑖 } �, (1) ˜𝑜𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 = 𝑚𝑖𝑛�{𝑙𝑎𝑡𝑜𝑖 𝑗 }𝑗 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=',𝑚𝑖 } �, (2) ˜𝑜𝑟𝑖𝑔ℎ𝑡 𝑖 = 𝑚𝑎𝑥 �{𝑙𝑛𝑔𝑜𝑖 𝑗 }𝑗 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=',𝑚𝑖 } �, (3) ˜𝑜𝑡𝑜𝑝 𝑖 = 𝑚𝑎𝑥 �{𝑙𝑎𝑡𝑜𝑖 𝑗 }𝑗 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=',𝑚𝑖 } �, (4) where 𝑙𝑛𝑔 denotes longitude and 𝑙𝑎𝑡 denotes latitude of 𝑙𝑜𝑖 𝑗 for sim- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The relative position 𝑟𝑝 𝑖 = {𝑟𝑝𝑙𝑒𝑓 𝑡 𝑖 ,𝑟𝑝𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 ,𝑟𝑝𝑟𝑖𝑔ℎ𝑡 𝑖 ,𝑟𝑝𝑡𝑜𝑝 𝑖 } is then calculated by the normalized distances between 𝑙𝑝𝑞 and each side of the ˜𝑜𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For example, 𝑟𝑝𝑙𝑒𝑓 𝑡 𝑖 is calculated as: 𝑟𝑝𝑙𝑒𝑓 𝑡 𝑖 = 𝑠𝑔𝑛(𝑙𝑛𝑔𝑝𝑞 − ˜𝑜𝑙𝑒𝑓 𝑡 𝑖 ) ∗ 𝑚𝑖𝑛�𝑘, ⌊𝑘 |𝑙𝑛𝑔𝑝𝑞 − ˜𝑜𝑙𝑒𝑓 𝑡 𝑖 | ˜𝑜𝑟𝑖𝑔ℎ𝑡 𝑖 − ˜𝑜𝑙𝑒𝑓 𝑡 𝑖 ⌋� + 𝑘, (5) where 𝑠𝑔𝑛(·) is the sign function, and ⌊·⌋ is the floor function that outputs the greatest integer less than or equal to a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 𝑘 ∈ N is a discretization factor that maps the relative distance ratio to a discrete number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As a result, we have 𝑟𝑝𝑙𝑒𝑓 𝑡 𝑖 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' , 2𝑘}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The discretized relative position feature is then encoded as e𝑝 𝑖 = {e𝑝𝑙𝑒𝑓 𝑡 𝑖 , e𝑝𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 , e𝑝𝑟𝑖𝑔ℎ𝑡 𝑖 , e𝑝𝑡𝑜𝑝 𝑖 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To extract the intrinsic features of geographic objects, the OSM IDs are mapped to embeddings in a similar way to word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The shape type 𝑜𝑠 is also mapped to embeddings like relation type 𝑟𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The ID embeddings of 𝑜𝑖 are denoted as e𝑑 𝑖 and its shape type embeddings as e𝑠 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Furthermore, to recognize the relationships among geographic objects, such as overlapping, the entire map area as a rectangle is split into a 𝑁 × 𝑁 grid to obtain its scale factors 𝑠𝑙𝑛𝑔 and 𝑠𝑙𝑎𝑡 for longitude and latitude respectively: 𝑠𝑙𝑛𝑔 = 𝑙𝑛𝑔𝑚𝑟𝑖𝑔ℎ𝑡 − 𝑙𝑛𝑔𝑚𝑙𝑒𝑓 𝑡 𝑁 ,𝑠𝑙𝑎𝑡 = 𝑙𝑎𝑡𝑚𝑡𝑜𝑝 − 𝑙𝑎𝑡𝑚𝑏𝑜𝑡𝑡𝑜𝑚 𝑁 , (6) where 𝑙𝑛𝑔𝑚𝑟𝑖𝑔ℎ𝑡 denotes longitude of the map’s right side and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The position of 𝑜𝑖 in the map 𝑜𝑚 𝑖 can thus be calculate with the scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For example, 𝑜𝑚𝑙𝑒𝑓 𝑡 𝑖 and 𝑜𝑚𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 are calculated as: 𝑜𝑚𝑙𝑒𝑓 𝑡 𝑖 = ⌊ ˜𝑜𝑙𝑒𝑓 𝑡 𝑖 −𝑙𝑛𝑔𝑚𝑙𝑒𝑓 𝑡 𝑠𝑙𝑛𝑔 ⌋ ∈ N, (7) 𝑜𝑚𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 = ⌊ ˜𝑜𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 −𝑙𝑎𝑡𝑚𝑏𝑜𝑡𝑡𝑜𝑚 𝑠𝑙𝑎𝑡 ⌋ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (8) The discretized position feature of 𝑜𝑖 in the map is then encoded as e𝑚 𝑖 = {e𝑚𝑙𝑒𝑓 𝑡 𝑖 , e𝑚𝑏𝑜𝑡𝑡𝑜𝑚 𝑖 , e𝑚𝑟𝑖𝑔ℎ𝑡 𝑖 , e𝑚𝑡𝑜𝑝 𝑖 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Finally, the geographic encoder sums these features of 𝑜𝑖 up as: e𝑖 = e𝑡 𝑖 + e𝑑 𝑖 + e𝑠 𝑖 + ∑︁ e𝑝 𝑖 + ∑︁ e𝑚 𝑖 (9) The intrinsic characteristics of geographic objects are described by the three components (e𝑑, e𝑠, and e𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' e𝑑 is the unique identifier of a geographic object, e𝑠 distinguishes road from AOI, e𝑚 depicts the positional relation among different geographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The other two components (e𝑡 and e𝑝) describe correlations between geolocation and geographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' After encoding surrounding geographic objects as a sequence {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' , e𝑚}, geographic encoder employs multi-layer bidirectional transformers [34] to learn inter- actions among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Following previous work [32], a 𝐺𝐶 token is prepended at the beginning like the 𝐶𝐿𝑆 token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The outputs of geographic encoder are therefore denoted as {h𝐺𝐶, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' , h𝑚}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geolocation l GCL Loss MGM Loss Single-Modal MLM Loss Multi-Modal MLM Loss Multi-Modal MGM Loss (106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='458,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='563) (Geo Input) (Geo Input) (Text Input) (Text & Geo Input) (Text & Geo Input) MASK el hcLs h y htm hsEP ei Add & Norm MASK hGC 01 Oi On eGC Geographic Objects en Feed-Forward e1 h1 Multi-Modal Interaction × Multi-Layer Add & Norm et hi e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ei Multi-Head Attention [pq ei CLS t1 tj tm SEP h1 ni el hn nn Relation: COVERED en Word Embeddings Geographic Encoder OSMID: 3119 Geographic Shape: POLYGON Encoder Text t Geolocation lA Multi-Modal Geographic Pre-Training Method Preprint, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We design two tasks to train geographic encoder and it is fixed in later uses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', masked geographic modeling (MGM) and geographic contrastive learning (GCL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Like the widely use masked language modeling (MLM) [5], MGM aims at predicting masked geographic features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', OSM IDs, geometric types, each side of the substituted rectangle, relation types, and relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The MGM loss 𝐿𝑀𝐺𝑀 is calculated by summing up the masked loss of all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' GCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' This task is related to multiple geolocations {𝑙𝑝𝑞 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑙𝑝𝑞 𝑏𝑠 } in a batch of size 𝑏𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We begin with the definition of the real-world geographic distance matrix H ∈ R𝑏𝑠×𝑏𝑠 defined as: H𝑖,𝑗 = 𝜎 �−∥ℎ𝑎𝑣𝑒𝑟𝑠𝑖𝑛𝑒(𝑙𝑝𝑞 𝑖 ,𝑙𝑝𝑞 𝑗 )∥N �,𝑖, 𝑗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ,𝑏𝑠},𝑖 ≠ 𝑗, (10) whereℎ𝑎𝑣𝑒𝑟𝑠𝑖𝑛𝑒 is the haversine function [23] that calculates spher- ical distance between geolocations, ∥ · ∥N is gaussian normalization function, and 𝜎 is sigmoid function that maps distance to range (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As the latent distance between embeddings in the output space should correspond to their real-world geographic distance, we use ℎ𝐺𝐶 as the representation of geolocation 𝑙𝑝𝑞 with GC and calculate the latent distance matrix ˜H ∈ R𝑏𝑠×𝑏𝑠 as: ˜H𝑖,𝑗 = ⟨∥h𝑖 𝐺𝐶 ∥𝐿2, ∥h𝑗 𝐺𝐶 ∥𝐿2⟩ (11) where ⟨·⟩ denotes the doc-product function and ∥ · ∥𝐿2 is 𝐿2 normal- ization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use KL-divergence to measure the similarity between H and ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' GCL loss 𝐿𝐺𝐶𝐿 is then calculated by: 𝐿𝐺𝐶𝐿 = 𝑏𝑠 ∑︁ 𝑖=1 𝐷𝐾𝐿 �𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(H𝑖) ∥ 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥( ˜H𝑖)� (12) where 𝐷𝐾𝐿(· ∥ ·) denotes the KL-divergence, and the 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥 function is applied to transform H𝑖 and ˜H𝑖 to a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The training loss 𝐿𝑔 of geographic encoder is thus calculated by: 𝐿𝑔 = 𝐿𝑀𝐺𝑀 + 𝐿𝐺𝐶𝐿 (13) Using such an training process, geographic encoder is capable of modeling GC in a given GIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Multi-Modal Pre-Training The input of MGeo pre-training is a pair of text and geolocation (𝑡, 𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The pre-training data can come from diverse sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', click of users or position of delivery clerks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The multi-modal training aims at aligning these two modals into one latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Word embeddings are used to map text into a sequence of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The geographic encoder provides the GC embeddings given 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The two embeddings are then concatenated together and fed into multi-layer bidirectional Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use three tasks to learn interaction between GC and text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', single-modal MLM, multi-modal MLM, and multi-modal MGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' These tasks are trained in turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Single-modal MLM is the original MLM task used in BERT, which randomly masks and replaces the input text with 𝑀𝐴𝑆𝐾 token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The outputs of geographic encoder are removed for single-modal MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' While multi-modal MLM predicts the masked token relying on the entire GC and part of textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Multi-modal MGM randomly masks and replaces the input geographic features with 𝑀𝐴𝑆𝐾 and predicts them relying on entire textual information and part of GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (A) Bi-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (B) Cross-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 3: MGeo can use both (A) bi-encoder and (B) cross- encoder architectures to measure relevance between query and POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Dashed line indicates that geolocation of query is optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ⊕ denotes element-wise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Relevance Measurement MGeo can use both bi-encoder and cross-encoder architectures, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-encoder encodes query and POI separately for efficiency issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' It can be used in both retrieval and ranking phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In practice, the GC of a POI or query is encoded by geo- graphic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since user location is not always available due to privacy issues or limited hardware, the GC of query can be ab- sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The outputs are then concatenated with word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Transformer-based multi-modal interaction module then produces hidden states as final representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We compute the similarity score of a query and POI pair by the cosine similarity between their 𝐶𝐿𝑆 representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', ˆh𝑝 and ˆh𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-encoder calculates similarity scores between a query and all the POIs for retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Different from bi-encoder, cross-encoder concatenates every query-POI pair together before being fed to multi-modal interaction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Cross-encoder allows fine-grained token-level interaction between query and POI, it usually provides a more accurate esti- mation of relevance but is less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Therefore, cross-encoder is only used in ranking phase as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The GC of query or POI is encoded separately by geographic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The GC of query is also optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We concatenate query textual embeddings, POI textual embeddings, query GC embeddings (optional), and POI GC embeddings together, which are then fed to multi-modal interaction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Particularly, we use geographic discriminator to facilitate geographic comparison between GC of query and POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geographic discriminator adds embeddings to outputs of geographic encoder to distinguish query GC from POI GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Like the segment embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Similarity Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='CLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='MGeo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='MGeo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Multi-Modal Interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Multi-Modal Interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Word Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Geographic Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Word Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Geographic Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Query Text tq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Query Geolocation 1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='POI Text tp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='POIGeolocation[p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='OptionalSimilarity Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='MGeo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Multi-Modal Interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Word Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Geographic Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Geographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Geographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Segment 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Segment 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Discriminator1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Discriminator 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Query Text tq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='POI Text tp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='Query Geolocation [q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='POI Geolocation [p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='OptionalPreprint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Table 2: Statistics of different query types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Query Type # Query Address 81,286 Street No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6,013 Colloquial 2,701 Total 90,000 Table 3: Statistics of train/dev/test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' # Query # Candidate POI Train 50,000 20 Dev 20,000 40 Test (Ranking) 20,000 40 Test (Retrieval) 2,849,754 Table 4: Statistics of geographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Line Polygon Covered Nearby Covered Nearby Query 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='005 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 POI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='003 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='4 in BERT, embeddings of geographic discriminator are randomly initialized and trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We fed the hidden states of 𝐶𝐿𝑆 ˆh𝑝𝑞 𝐶𝐿𝑆 to a multi-layer perceptron (MLP) to produce similarity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 5 THE GEOTES BENCHMARK In this section, we introduce our proposed large-scale benchmark GeoTES, which stands for Geographic TExtual Similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' It is the first open-source benchmark for query-POI matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The POIs are obtained from the open-source OSM and the queries are manually generated by annotators to prevent privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Annotation Process In this version of GeoTES, all the POIs are located in Hangzhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 20 annotators and 4 experienced experts are asked to annotate three types of queries based on the POIs, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Ta- ble 2 gives the statistics of these query types, which follows the distribution of our online LBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In OSM, each POI comes with a geographic location under the WGS84 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Neigh- bouring POIs of the OSM POIs from several open-accessed map services are selected by the annotators to enrich the diversity of POI description and also serve as hard negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To simulate the queries’ location in real scenes, the annotators are asked to ran- domly select a location within 1km of corresponding POI for 50% of the queries and randomly select a location in the city for the rest queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' All the annotators have adequate linguistic knowledge and educational/cultural background to produce appropriate queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To 3https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='org/wiki/Converting_to_WGS84 Table 5: Model sizes of pre-trained and fine-tuned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Pre-Train Fine-Tune BERT-DA 118M 102M BERT-MGeo 213M 129M eliminate biases during the annotation process, they are instructed with detailed annotation principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' One quality inspector ensures that each of the queries has one matched POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Benchmark Statistics GeoTES has a total number of 90,000 queries with an average length of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 and 2,849,754 POIs with an average length of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We extract the geographic surrounding objects for the queries and POIs from OSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' There are 21,950 lines and 65,722 polygons in our extracted geographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As given in table 4, each query and POI has more relations to polygons than lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The benchmark is randomly split in to train, development, and test sets, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For the train, development, and ranking test sets, we provide a list of candidate POIs and ensure that one exact matched POI is contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The retrieval test set use the same queries as the ranking test set while no candidate POI should be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Therefore, we believe that the GeoTES presents a reliable and challenging dataset for benchmarking retrieval and ranking models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6 EXPERIMENT In this section, we compare the proposed MGeo with several strong baselines on GeoTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The experiments are conducted on two tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', ranking and retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The two tasks use the same train, development, and test sets as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' A list of candidate POIs that contains the relevant one is provided for the ranking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Both bi-encoder and cross-encoder are evaluated on ranking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since retrieval task requires searching the full POI corpus, and cross-encoder needs too much computing resources to complete retrieval task, only bi- encoder is evaluated on retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Following previous IR work [24], we use Re- call and Mean Reciprocal Rank (MRR) at top 𝑘 ranks to evaluate the performance on both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Recall@𝑘 calculates the proportion of queries that have the relevant POI contained in the top-𝑘 candi- dates, and MRR@𝑘 calculates the averaged reciprocal of the rank at which the relevant POI is placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We report the evaluation scores on the test set of models that perform best on the development set during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' PTM Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We first evaluate the performance of four widely used PTMs with the base model size on GeoTES, including BERT [5], RoBERTa4 [21], ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0 [31], and StructBERT [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We further apply domain-adaptive pre-training techniques (DA) on BERT and another top-performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' DA is a widely used single-modal pre-training baseline [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For a fair comparison, domain corpus used in DA is the same as that used in our proposed multi-modal 4https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='co/clue/roberta_chinese_base A Multi-Modal Geographic Pre-Training Method Preprint, Table 6: Ranking results of bi-encoder and cross-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bold indicates the best of each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-Encoder Cross-Encoder PTM Recall@1 Recall@3 Recall@5 MRR@5 Recall@1 Recall@3 Recall@5 MRR@5 BERT 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='83 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='40 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='24 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='60 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='52 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='11 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='10 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='53 RoBERTa 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='52 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='41 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='25 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='15 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='20 93.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='96 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='51 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='21 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='67 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='53 BERT DA 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='49 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='18 93.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='78 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='06 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='28 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='65 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='33 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='92 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='61 BERT MGeo w/o query GC 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='86 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='61 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='53 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='93 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='11 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='42 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='75 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='86 StructBERT 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='37 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='96 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='89 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='72 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='85 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='16 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='40 BERT MGeo 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='04 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='24 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='18 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='85 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='89 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='48 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='48 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='74 StructBERT 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='07 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='68 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='50 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='57 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='49 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='55 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='62 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='10 Table 7: More ranking results of bi-encoder and cross- encoder baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Recall@1 Bi-Encoder DSSM [39] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='59 DPAM [39] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='15 PALM [39] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='51 BERT 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='83 ColBERT [15] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='36 Poly-Encoder [13] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='87 BERT-MGeo 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='04 Cross-Encoder BERT 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='52 ERNIE-GeoL [11] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='94 BERT-MGeo 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='89 geographic pre-training (MGeo), except that MGeo has additional GC along with query and POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Hyperparameter The architecture of the multi-modal interaction module is multi- layer transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The model sizes are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Geographic Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' All geographic feature embeddings are set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The discretization factor 𝑘 is 10 and the grid number 𝑁 is 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Geographic encoder has 4 layers of transformer with 256 hidden sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The mask probability is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The training batch size is 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use AdamW as optimizer with learning rate being 1e-4, weight decay being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We train geographic encoder for 30 epochs and take the last epoch checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Pre-Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The training batch size is 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use AdamW as optimizer with learning rate being 5e-5, weight decay being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We train for 10 epochs and take the last epoch checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Downstream Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For bi-encoder models, every training step has 56 queries, each has 20 candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use AdamW as optimizer with learning rate being 5e-5, weight decay being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Specifically, Table 8: Retrieval results of bi-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' BERT BERT-DA BERT-MGeo Recall@1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='70 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='76 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='70 Recall@3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='06 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='36 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='28 Recall@5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='32 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='82 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='39 Recall@20 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='70 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='08 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='49 Recall@50 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='30 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='61 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='00 Recall@100 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='02 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='74 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='29 MRR@5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='58 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='29 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='79 MRR@10 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='98 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='71 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='25 ERNIE and StructBERT don’t converge in this learning rate, we change it to 5e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We train geographic encoder for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For cross-encoder models, every training step has 24 queries and the learning rate for RoBERTa is 5e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Other settings are the same as bi-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Ranking Table 6 gives the ranking results of both bi-encoder and cross- encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As the original StructBERT outperforms the other generic PTMs, it is used for further DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The generic PTMs directly fine- tuned on the downstream tasks show a low performance, which indicates that these two tasks are challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since cross-encoder can make fine-grained interactions among input features, while bi-encoder only interacts with the 𝐶𝐿𝑆 representations for the sake of efficiency, cross-encoder generally outperforms bi-encoder by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' By applying DA on bi-encoder, PTMs could gain an advantage over the generic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' However, DA models consider only the tex- tual modality and neglect the other geographic modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Through multi-modal pre-training, MGeo without query GC raises 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='37% (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='07%) point of Recall@1 on BERT-DA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', StructBERT-DA) by bridging the gap between query text and POI GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' After being accompanied by query GC, MGeo further shows a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='55% (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='77%) improvement in Recall@1 over DA models with the help of incorporating correlations between query GC and POI text, as well Preprint, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Table 9: Inference time (second) of bi-encoder and cross- encoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-Encoder Cross-Encoder BERT-DA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0396 BERT-MGeo w/o query GC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0414 BERT-MGeo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='0466 as between query GC and POI GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' It is worth noting that GC of half the training and test queries are noises to simulate the arbitrary geolocation of users, as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The results show MGeo’s capability of denoising and it may gain more improvements if the queries have more precise geolocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In cross-encoder, MGeo also shows superiority over baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' DA brings fewer benefits on PTMs than it does in bi-encoder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='72% on BERT and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='14% on StructBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' However, improvements brought by incorporating the new geographic modal are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' MGeo without query GC gains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='87% (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='07%) Recall@1 on BERT-DA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', StructBERT-DA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Together with query GC, MGeo boost DA models by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='65% (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='84%) in Recall@1, showing effectiveness of multi-modal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 More Baseline Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Besides the PTM baselines, we also add more query-POI matching baselines, including two SOTA text-matching models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', ColBERT [15] and Poly-Encoder [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ColBERT uses a late interaction architecture to enhance bi-encoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Similarly, Poly-Encoder uses attention mechanism to capture richer interactions between query and POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Detailed introductions of DSSM, DPAM, and PALM can be found in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ERNIE-GeoL is a strong PTM cross-encoder baseline introduced in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since the data and code of ERNIE-GeoL are not released, we only adopt the pre-training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The results on the ranking task are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For bi-encoder, BERT-MGeo still outperforms ColBERT and Poly-Encoder, which capture more fine-grained interactions between query and POI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For cross-encoder, ERNIE-GeoL uses spe- cific pre-training objectives to capture static geographic knowledge and outperforms BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' While BERT-MGeo capture dynamic GC and outperforms ERNIE-GeoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Retrieval Bi-encoder is also evaluated on retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since retrieval task focuses on finding the relevant POIs rather than ranking the correct POI at the top, table 8 reports Recall and MRR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Compared to BERT-DA, MGeo improves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='41% Recall@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The results demonstrate that the effectiveness of MGeo in bi-encoder architecture stays consistent when the size of candidates becomes 100,000 times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='4 Inference Time The inference time on 1 NVIDIA V100 GPU of bi-encoder and cross- encoder models is listed in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For bi-encoder, we only count the time of query encoding, since the document can be encoded in advance in many industrial scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We use 26 queries and 1040 documents for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Table 10: Influence of different geographic object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-Encoder Cross-Encoder Recall@1 MRR@5 Recall@1 MRR@5 Line 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='56 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='57 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='71 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='88 Polygon 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='26 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='51 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='84 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='85 Both 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='04 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='85 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='89 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='74 Figure 4: Ranking MRR@5 for different percentage of query with GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='5 Ablation Study Since we use the same bi-encoder models for both retrieval and ranking tasks, the ablation study is mainly conducted on ranking task of BERT-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='1 Geographic Object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We first study the influence of training queries with GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We randomly remove GC of the same proportion from the training, development, and test queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As shown in Figure 4, the performance is impaired when a small proportion of queries contain GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' This decrease comes from a larger proportion of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Taking 30% of queries having GC as example, there are already 15% GC are noises (half GC are randomly selected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Since it is difficult to distinguish query without GC from query without geographic object (but with geolocation), the rest queries without GC can be considered as noises too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Thus we have in total 75% queries with noisy GC, which damages model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' When noises proportion becomes smaller than 65% (70% query with GC), the performance is better than training without query GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The influence of different geographic object types is reported in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' There is not a huge gap between line and polygon for bi- encoder, while cross-encoder can perform better with only polygon than only line, as there are more polygons than lines in the GIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' This also suggests that cross-encoder is better at capturing the fine- grained correlations than bi-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Nevertheless, using either line or polygon is better than the single-modal baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Besides, Bi-Encoder Cross-EncoderA Multi-Modal Geographic Pre-Training Method Preprint, (A) Bi-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (B) Cross-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 5: Ranking Recall@1 for (A) bi-encoder and (B) cross- encoder with different query types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' bi-encoder and cross-encoder can have a better performance when the two types of geographic objects both present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='2 Query Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 5 shows the performance on three query types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=', address, street number, and colloquial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Bi-encoder models perform best on address description, while cross-encoder models perform best on street number description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' This suggests that cross- encoder is better at capturing fine-grained correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Colloquial query contains many daily expressions, which rarely appear in domain corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Thus BERT-DA is even worse than BERT on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' However, the use of GC help reduce this shortcoming of DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='3 Amount of Training Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We study the performance of MGeo with different amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' As shown in Figure 6, the dashed line is used for representing BERT-DA and the dotted line for the original BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' With only 30% of training data, the bi-encoder and cross-encoder using MGeo can outperform the BERT baseline by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content='4 Query Incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' POI suggestion also plays an impor- tant role in LBS, where the name of POIs are listed when the input (A) Bi-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (B) Cross-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 6: Ranking Recall@1 for (A) bi-encoder and (B) cross- encoder with different amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (A) Bi-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' (B) Cross-Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 7: Ranking Recall@1 for (A) bi-encoder and (B) cross- encoder with different percentage of query incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' is unfinished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' To simulate such scenario, we also evaluate MGeo on incomplete queries by truncating the trailing characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Figure 7 shows the performance with different truncation ratio of the test queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' The results demonstrate that bi-encoder using MGeo could outperform the BERT baseline with full queries with a small truncation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Whereas the cross-encoder could not, since the semantic similarity is more important for cross-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' 7 CONCLUSION In this paper, we formalize an important concept GC, which is indispensable for real-world human POI exploration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' We propose a multi-modal geographic language model MGeo, which considers GC as a new modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Therefore, GC can be represented to- gether with text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' In addition, we build a new open-source large-scale benchmark GeoTES to facilitate further research on the query-POI matching topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Extensive experiments are conducted to evaluate our proposed method on the state-of-the-art PTMs, and the detailed analyses demonstrate that MGeo can significantly outperform other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Even though geolocation of user may be absent and query has no GC, MGeo can still obtain improvements over the baselines, showing its capability of modeling text-to-text, GC-to-GC and text- to-GC correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' For future work, other modalities like POI image can be further explored, as well as more inventive geographic en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' Besides, our proposed GC modeling has the potential to boost all geography-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' BERT-MGeo BERT-DA BERTBERT-MGeo BERT-DA BERTAddress Street No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ColloquialAddress Street No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' ColloquialBERT-MGeo BERT-DA BERTBERT-MGeo BERT-DA BERTPreprint, Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfDAny/content/2301.04283v1.pdf'} +page_content=' REFERENCES [1] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexan- 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b/RNFJT4oBgHgl3EQfKSyK/content/tmp_files/2301.11464v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b6295f682e7a82fe740d5b499b3cb0080c4fdeb --- /dev/null +++ b/RNFJT4oBgHgl3EQfKSyK/content/tmp_files/2301.11464v1.pdf.txt @@ -0,0 +1,1480 @@ +MNRAS 000, 1–13 (2023) +Preprint 30 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The formation of supermassive black holes from Population III.1 seeds. II. +Evolution to the local universe +Jasbir Singh,1,2,3,4★ Pierluigi Monaco,1,2,3,5 Jonathan C. Tan4,6 +1Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy +2INAF- Astronomical Observatory of Trieste, via Tiepolo 11, 34143 Trieste, Italy +3IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34014 Trieste, Italy +4Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden +5INFN, Sezione di Trieste, 34149 Trieste, Italy +6Dept. of Astronomy, University of Virginia, Charlottesville, VA 22904, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present predictions for cosmic evolution of populations of supermassive black holes (SMBHs) forming from Population +III.1 seeds, i.e., early, metal-free dark matter minihalos forming far from other sources, parameterized by isolation distance, 𝑑iso. +Extending previous work that explored this scenario to 𝑧 = 10, we follow evolution of a (60 Mpc)3 volume to 𝑧 = 0. We focus +on evolution of SMBH comoving number densities, halo occupation fractions, angular clustering and 3D clustering, exploring +a range of 𝑑iso constrained by observed local number densities of SMBHs. We also compute synthetic projected observational +fields, in particular a case comparable to the Hubble Ultra Deep Field. We compare Pop III.1 seeding to a simple halo mass +threshold model, commonly adopted in cosmological simulations of galaxy formation. Major predictions of the Pop III.1 model +include that all SMBHs form by 𝑧 ∼ 25, after which their comoving number densities are near-constant, with low merger rates. +Occupation fractions evolve to concentrate SMBHs in the most massive halos by 𝑧 = 0, but with rare cases in halos down to +∼ 108 𝑀⊙. The 𝑑iso scale at epoch of formation, e.g., 100kpc-proper at 𝑧 ∼ 30, i.e., ∼ 3Mpc-comoving, is imprinted in the SMBH +two-point angular correlation function, remaining discernible as a low-amplitude feature to 𝑧 ∼ 1. The SMBH 3D two-point +correlation function at 𝑧 = 0 also shows lower amplitude compared to equivalently massive halos. We discuss prospects for +testing these predictions with observational surveys of SMBH populations. +Key words: black holes – formation – early universe +1 INTRODUCTION +The formation of stellar mass black holes is relatively well under- +stood, but the same is not true for supermassive black holes (SMBHs). +These black holes have masses ≥ 105𝑀⊙ and are found at the center +of most large galaxies. The biggest mystery regarding their formation +is explaining their high masses in the early universe. A stellar-mass +BH, formed at very high redshift from the collapse of a massive +primordial star, can grow by accreting gas as long as accretion can +be sustained for a long time. However, this accretion is believed to +be Eddington-limited by radiation pressure, so when gas inflow is +abundant the growth of BH mass is expected to be exponential, with +an e-fold time of ∼ 4 × 107 yr. +Recent discoveries of high redshift quasars, for example J0313- +1806 at 𝑧 = 7.642 (farthest observed to date, Wang et al. 2021) and +J1007+2115 at 𝑧 = 7.515 (Yang et al. 2020), both hosting a SMBH +more massive than 109𝑀⊙ put stringent constrain on any SMBH +formation scenario. The existence of these quasars imply that these +black holes grew to such high masses by the time the universe was +only ∼ 700 million years old. Even assuming a very early formation +at 𝑧 ∼ 30, the BH seed should be at least as massive as 500 𝑀⊙ +★ E-mail: jasbir.singh@inaf.it +to grow to the desired mass by the observation redshift, and later +formation would imply higher seed masses. +A variety of theories have been proposed to explain the formation +of SMBHs, with different degrees of complexity and rooted to small- +scale physics that is typically unresolved a cosmological simulations. +As a consequence, simplified assumptions are typically used in these +simulations to create black holes in a given dark matter halo or the +galaxy contained in it, based on the properties of the parent halo +or the galaxy, often using sub-grid physics. One of the simplest +and widely used models is the halo mass threshold (HMT) seeding +scheme based on the methods developed by Sijacki et al. (2007) and +Di Matteo et al. (2008), in which a seed black hole is assumed to +form in a halo crossing a certain mass threshold. The Illustris project +(Vogelsberger et al. 2014) uses this mechanism to add SMBHs of +mass 1.4 × 105𝑀⊙ in each halo which crosses a mass threshold of +𝑚th = 7.1×1010𝑀⊙. A similar approach is used in the Evolution and +Assembly of GaLaxies and their Environments (EAGLE) simulation +(Barber et al. 2016). +There have been many attempts to explain the formation of SMBHs +via more physical mechanisms, dating back to the last century (e.g., +Rees 1978). One of the most popular mechanisms is direct collapse, +which involves the collapse of a large primordial composition gas +cloud in a halo of mass ∼ 108𝑀⊙ into a single supermassive star +© 2023 The Authors +arXiv:2301.11464v1 [astro-ph.GA] 26 Jan 2023 + +2 +Singh et al. +of 104 − 106𝑀⊙ that then collapses to form a SMBH at the centre +of the halo (Bromm & Loeb 2003; Begelman et al. 2006; Lodato & +Natarajan 2006; Shang et al. 2010; Montero et al. 2012). Although the +number density of black holes emerging from direct collapse would +be enough to explain the currently known population of high redshift +quasars, the conditions required for this scenario are not thought to be +common enough to explain the total observed population of SMBHs +at 𝑧 = 0 (Chon et al. 2016; Wise et al. 2019). Furthermore, recent +simulations have shown that the supermassive stars forming via this +mechanism might not be as massive as initially predicted, but only +reaching ≲ 104𝑀⊙, due to the turbulent environment present in the +initial stages of galaxy formation, which disrupts the accretion flow +(Regan et al. 2020). +Another mechanism to form intermediate, or even supermassive +black holes is through runaway stellar mergers in young and dense +clusters to create massive stars as seeds of the order ∼ 200 − 103𝑀⊙ +(e.g., Portegies Zwart et al. 2004). This mass can be reached through +repeated collisions if the massive stars can reach the cluster core to +increase the collision rate drastically (Ebisuzaki 2003) before they +explode as supernovae. However, predicting whether such conditions +arise in galaxies and at what rate is very challenging given the the +need to resolve the formation and evolution of individual stars, so pre- +dictions for the cosmological population of such systems are highly +uncertain (see, e.g., Boekholt et al. 2018; Chon & Omukai 2020; +Tagawa et al. 2020). +Some methods take into consideration more local properties of +the host galaxy, such as the ones used in Horizon-AGN simulation +(Volonteri et al. 2016), in which the gas and stellar densities and the +stellar velocity dispersion is required to exceed a certain threshold +for the galaxy to be seeded with a black hole. In addition to this, all +the forming black holes must be separated by at least 50 comoving +kpc, and the formation is limited until 𝑧 = 1.5. If all these conditions +are met, the halo is seeded with a 105𝑀⊙ black hole. Adopting a +similar criteria, the more recent obelisk simulation (Trebitsch et al. +2021) also applies the condition of gas and stellar density exceeding +a threshold, and an isolation of 50 kpc to avoid multiple black holes +forming in the same galaxy. Furthermore, they also require the gas +to be Jeans unstable. If all these conditions are satisfied, then a black +hole of 3 × 104𝑀⊙ is assigned to the galaxy. In another approach +that also uses the local properties of the galaxy to assign a seed, +the romulus simulation (Tremmel et al. 2017) employs the criteria +of the limit on the metallicity, a threshold on the gas density, and a +temperature range. Once all these conditions are satisfied, the mass +of the seed black hole is set to be 106𝑀⊙. +In this work, we focus on a formation scenario which invokes the +Population III.1 stars formed in the early universe as the progenitors +of SMBHs. Pop III.1 stars are defined to be Pop III (i.e., metal +free) stars forming in first dark matter minihalos that are isolated +from other stellar or SMBH feedback sources (McKee & Tan 2008). +It is assumed that in the absence of any significant radiative (or +mechanical) feedback, a single dominant protostar forms at the center +of the minihalo and has its structure affected by the energy input +from Weakly Interacting Massive Particle (WIMP) dark matter self +annihilation inside the protostar (Spolyar et al. 2008; Natarajan et al. +2009; Freese et al. 2010; Rindler-Daller et al. 2015). Such protostars +maintain relatively cool outer layers, which allows efficient accretion +of the baryonic content of the minihalo, i.e., ∼ 105 𝑀⊙, to form +a supermassive star, which subsequently collapses efficiently to a +SMBH after a few Myr. +This Pop III.1 seeding mechanism, which is based on locating +isolated minihalos, was applied on a cosmological simulation in +Banik et al. (2019) (hereafter Paper I). The evolution was followed +from high redshifts down to 𝑧 = 10. The main free parameter in the +model is the isolation distance (𝑑iso), i.e., how far a newly forming +minihalo needs to be from previously formed halos in order to be +a Pop III.1 source. For a fiducial value of 𝑑iso = 100 kpc (proper +distance), the model yields co-moving number densities of SMBHs +that match the estimated level of the known 𝑧 = 0 SMBH population. +Note, that in this case (and all other reasonable cases) most minihalos +do not form Pop III.1 sources. Rather, most are Pop III.2 sources, +which are metal free, but having been disturbed by radiative feedback +undergo significant fragmentation to form only lower-mass (e.g., +∼ 10 𝑀⊙) stars (Greif & Bromm 2006). +In this paper, we take this Pop III.1 seeding mechanism and extend +the results down to the local universe, 𝑧 = 0. In §2, we briefly describe +our seeding algorithm and the tools used to apply it. Then we present +our results in §3, starting with the evolution of number density of +seeded halos down to 𝑧 = 0. We compare these results with the HMT +scheme, and also discuss the SMBH occupation fraction and cluster- +ing properties of seeded halos. Finally, we create synthetic Hubble +Ultra Deep Fields (HUDFs) to demonstrate the possibility of using +the HUDF to differentiate among different seeding mechanisms. We +then present our conclusions in §4. +2 METHODS +2.1 pinocchio simulations +As in Paper I, to test our Pop III.1 seeding mechanism, we used +the Pinocchio code (Monaco et al. 2002; Munari et al. 2017) to +generate a cosmological box of 59.7 Mpc (40 ℎ−1 Mpc for ℎ = +0.67) with standard Planck cosmology (Planck Collaboration 2020) +and study the formation of DM (mini-)halos in that box. Pinocchio +uses Lagrangian Perturbation Theory (LPT, e.g., Moutarde et al. +1991) to approximate the evolution of cosmological perturbations +in a ΛCDM universe. For a given set of initial conditions, the code +generates outputs in the form of catalogs at different redshifts, which +contain mass, position and velocity of the DM halos, and a complete +information of the merger histories of all the halos, with continuous +time sampling. +This code was written for applications in cosmology, where huge +volumes with moderate mass resolution are requested, and its perfor- +mance heavily depends on the mass resolution adopted. To resolve +minihalos of ∼ 106𝑀⊙ it is necessary to sample a 59.7 Mpc box +with 40963 particles; this results in a particle mass of 1.23×105𝑀⊙, +and we adopted a minimum mass of 10 particles (that would be +unacceptable for an N-body simulation, but it is acceptable for a +semi-analytic code like Pinocchio), resulting in a minihalo mass of +1.23 × 106𝑀⊙. Such a large simulation can only be run on a super- +computer, distributing the computation on a large number of nodes. +Since the fragmentation of collapsed particles into halos is done in +Lagrangian space, the domain distributed to a task is not much larger +than the dimension of the largest halo, so massive halos will not be +reconstructed correctly. As a result, with V4 of pinocchio (Munari +et al. 2017) used in Paper I, we were only able to push the simulation +down to 𝑧 = 10. +We use here the novel V5 of the code, that implements a number of +numerical techniques to improve memory efficiency. This code will +be presented elsewhere, the strategy to perform halo construction +at high resolution is the following: a first step of halo construction +is performed using subboxes; then the domain is augmented with +all particles that lie within 𝑁Lag times the Lagrangian size of the +constructed halos; and then halo construction is performed again. +MNRAS 000, 1–13 (2023) + +SMBH formation and evolution to the local universe +3 +Memory occupation depends on 𝑁Lag, so we were forced to use +𝑁Lag = 2, while a value of 3 is a better guarantee of convergence +in halo construction. The 59.7 Mpc box with full 40963 resolution +was run to z=0 on 800 MPI tasks over 100 computing nodes (each +with 256 GB of RAM), so the domain was divided into 6 × 6 × 7.5 +Mpc sub-volumes for halo construction. The resulting halo mass +function showed two problems that are presented in greater detail in +an Appendix. We discuss here their nature and their implications. +As a consequence of the difficulty of calibrating the formation of +halos with a very steep power spectrum, the mass of the first halos is +underestimated by a factor of ∼ 2 at 𝑧 ∼ 30, decreasing to a negligible +value at 𝑧 ∼ 10. This is a known trend in pinocchio, visible, e.g., +in Figure 1 of Munari et al. (2017) where the 𝑧 = 3 halo MF is +slightly underestimated in those tests. We are working to improve this +prediction, but we do not consider this as a showstopper for several +reasons: our seed BHs are already predicted to form very early, so +this underestimation only causes us to be slightly conservative in +their formation redshift, i.e., in fact they would already have formed +at slightly higher 𝑧. In our simple modeling we are assuming here +immediate formation of the protostar and then the SMBH, whereas +in reality this might take several Myr or even tens of Myr. The +time span that separating 𝑧 = 32 from 𝑧 = 29 is only ∼ 14 Myr, +so neglecting astrophysical timescales leads to an overestimation of +formation redshift, which compensates against the underestimation +problem. Finally, the minihalo threshold mass can be consider to be a +second free parameter of the modeling (although one that has physical +motivation to be close to 106 𝑀⊙), so one can simply consider our +predictions to be valid for minihalo masses of 2.5×106 𝑀⊙. We add +to these arguments the fact that inaccuracies in halo masses do not +propagate as inaccuracies in halo positions, that are crucial outcomes +of our seeding scheme. +A more serious problem is connected to the inaccurate reconstruc- +tion of halos more massive than 1012𝑀⊙. Indeed, the small size of +the sub-box domain for constructing halos results in a poor recon- +struction of massive halos. This problems makes predictions at 𝑧 = 0 +unreliable. We thus produced the same box at a lower resolution, +sampled with 10243 particles, on a single MPI task on a 256 GB +node. Again, this was possible thanks to V5 of the code. In this case +halo construction is as good as it can be. However, the identification +of halos that contain seed SMBHs has been performed in the high +resolution box, and though the simulations share the same large-scale +structure, matching massive halos in the two boxes is not a clean pro- +cedure. We then resorted to this algorithm: starting from the fact that +one low-resolution particle contains 64 high-resolution ones, we cal- +culated which particle in the lower resolution box includes the seeded +mini-halo, and assigned the seed to the halo that contains that specific +low-resolution particle. We checked that results at 𝑧 = 0 produced +with the low- and high-resolution simulations were consistent, with +a significant difference in halo clustering of halos more massive than +a certain threshold that is an expected consequence of the inaccurate +mass reconstruction and the known relation of halo bias with halo +mass. In the following we will present results at 𝑧 = 0 based on the +low resolution box, unless mentioned otherwise. +2.2 Seeding scheme +To determine which halos are seeded with a Pop III.1 star and thence +SMBH, consider the scenario depicted in Fig. 1, unfolding in the +early universe. The figure shows three stars A, B and C in different +halos where only A and C become Pop III.1 stars whereas B is a Pop +III.2 star, depending on the separation and formation order. Star A +Figure 1. A schematic illustration of the Pop III.1 SMBH seeding scenario +depicting the conditions for a star to be isolated enough to be considered as a +Pop III.1 star (see text). +formed first, which then influenced its environment within a sphere of +radius equal to 𝑑feedback, expected to be primarily radiative feedback. +Since this star is in a pristine primordial gas without the influence of +any feedback from nearby stars, it is defined to be a Pop III.1 star. +Star B, which subsequently forms at a distance less than 𝑑feedback +from star A, is affected by the feedback and hence is a Pop III.2 star +(or even a Pop II star if it has been chemically polluted). Finally, +star C forms outside the sphere of influence of both A and B, and +is thus also assigned to be a Pop III.1 star and thus a SMBH. For +the model considered here, the feedback distance is set equal to the +isolation distance 𝑑iso. So effectively, the condition for a star to be +regarded as a Pop III.1 star is that when it is forming, there should be +no previously formed halos present in the sphere of radius 𝑑iso. We +consider 𝑑iso as a free parameter in our theory and vary it to match +the observed number density of the SMBHs in the local Universe. +2.3 Seed identification in the dark matter catalogs +To perform the seed identification analysis from the dark matter cata- +logs generated by pinocchio, we first divided the entire redshift range +(from 𝑧 = 0 to the redshift when the first minihalo forms, 𝑧 ≈ 40) +into small bins of widths ranging from Δ𝑧 = 1, 2 or 3, depending on +the output catalogs available, which in turn depends on the relative +change in positions of (mini)halos. The bins are wider at high red- +shifts, but smaller at lower redshifts. Then for each redshift interval +(𝑧𝑙, 𝑧ℎ] where (𝑧ℎ > 𝑧𝑙), we utilised k-d tree data structure to create +a three dimensional map in position space of all the halos existing be- +tween 𝑧ℎ and 𝑧𝑙. The positions used to create the tree are taken from +the output catalog of pinocchio at the lower redshift of the interval +(𝑧𝑙). Since the positions are not updated once the tree is constructed, +we account for the change in the positions within this redshift interval +by finding the maximum change (𝛿) of position among all the halos +existing for the entire redshift range. Then for each minihalo crossing +the mass threshold of 106𝑀⊙ (or as in the nomenclature of pinoc- +chio: "appearing") at a redshift 𝑧app ∈ (𝑧𝑙, 𝑧ℎ], we perform a ball +search using the k-d tree to find all the halos around the appearing +minihalo within a sphere of radius 𝑑iso − 2𝛿1. If there exists even a +single halo at the redshift 𝑧app within this sphere, then this minihalo +1 A factor of 2 is multiplied with 𝛿 to account for the change in position of +both the minihalo at the center of the sphere and all the other halos within the +sphere. +MNRAS 000, 1–13 (2023) + +A: Pop Ill.1 +Afeedback = +★ +B: Pop Ill.2 +C: Pop Ill.14 +Singh et al. +is flagged as a halo containing a non-Pop III.1 star at its center. If +there are no halos existing at this redshift, then the ball search is +performed again with the same minihalo at the center, but this time +within a sphere of radius 𝑑iso + 2𝛿. Then for all the halos existing at +redshift 𝑧app within the shell of radius 𝑑iso ± 2𝛿, we find the exact +distance between the minihalo at the center and all these halos using +the exact positions at 𝑧app. If this distance is greater than 𝑑iso for all +the halos within the shell, then the minihalo at the center is flagged +as a Pop III.1 source, i.e., an SMBH-seeded halo. This process is +repeated for each minihalo crossing the threshold mass within the +two redshifts, and then this whole procedure is performed again for +all the redshift intervals, until the whole redshift range is covered. In +this way we are able check the isolation condition for each minihalo +appearing in the cosmological box and find all the seeded minihalos. +At smaller redshifts, the change in positions of the halos (𝛿) within +the redshift intervals becomes comparable to the isolation distance. +This implies that the quantity 𝑑iso − 2𝛿 can become negative (in our +simulation box, this happens at around 𝑧 ≈ 15 for 𝑑iso = 50 kpc). In +this case, the ball search is directly performed in a sphere of radius +𝑑iso + 2𝛿, and then the exact distances between the minihalo at the +center and all the other halos existing at 𝑧app is calculated. +This division of the entire redshift interval and creating the k-d +only at specific redshifts is performed to avoid reconstructing the +tree with the up-to-date position at every instance a new minihalo +appears. Since the number of minihalos is very large, it becomes +highly expensive computationally to reconstruct the tree with updated +positions each time a new minihalo appears. +3 RESULTS +3.1 Number density evolution +As explained in the last section and in detail in Paper I, we identify +SMBH-seeded halos by the condition that the isolation sphere of +radius 𝑑iso around a newly forming minihalo is not be populated by +any other existing halo (of mass greater than our minihalo threshold +mass). The obtained results for the evolution of number density for +different values of 𝑑iso (in proper distance units) are shown in Fig. 2. +The estimate for the observed number density of SMBHs in the local +Universe, 𝑛SMBH(𝑧 = 0) (black square in the figure) is calculated +by assuming that each galaxy with luminosity greater than 0.33𝐿∗ +hosts a SMBH (see Paper I). Here 𝐿∗ is the characteristic luminosity +corresponding to 𝑀B = −19.7 + 5 log ℎ = −20.55 (for e.g., Norberg +et al. 2002). The colored dotted lines show the number density evolu- +tion of total number of SMBHs, whereas the colored solid lines show +the number density for seeded halos (which can be slightly smaller +due to mergers). These results are from the highest resolution sim- +ulation with 40963 particles. Compared to the number densities in +Figure 1 of Paper I, the values obtained here are slightly lower (by +a factor of ∼ 1.45 for 100 kpc, and ∼ 1.65 for 50 kpc) because we +have considered periodic boundary conditions when identifying the +seeds, which was not done in Paper I. +From Fig. 2, it can be clearly seen that as the isolation distance is +reduced, the number of formed SMBHs increases. This is expected +because smaller 𝑑iso results in more halos satisfying the isolation +criteria for hosting SMBH seeds within our simulation volume. We +can also conclude that for a certain range of 𝑑iso (≈ 90 kpc to 170 +kpc), the number density obtained is in reasonable agreement with the +𝑧 = 0 estimate. A key feature of the fiducial Pop III.1 SMBH seeding +model, i.e., with 𝑑iso = 100 kpc, is that all SMBHs have formed very +early in the Universe: the process is essentially complete by 𝑧 ≃ 25. +Table 1. Total number of formed SMBHs (𝑁SMBH,form), total number of +SMBHs remaining at 𝑧 = 0 assuming efficient mergers (𝑁SMBH(𝑧 = 0)), the +difference between these (Δ𝑁SMBH = 𝑁SMBH,form − 𝑁SMBH(𝑧 = 0)), which +is equivalent to the number of mergers, and the fraction of original SMBHs +that are destroyed by mergers ( 𝑓merger = Δ𝑁SMBH/𝑁SMBH,form). +𝑑iso [kpc] +𝑁SMBH,form +𝑁SMBH(𝑧 = 0) +Δ𝑁SMBH +𝑓merger +50 +15470 +14499 +971 +6.28 +75 +3394 +3303 +91 +2.68 +100 +1234 +1222 +12 +0.97 +150 +306 +306 +0 +0 +200 +121 +121 +0 +0 +We compare this prediction to a halo mass threshold model (HMT +scheme; shown by the green dashed line in the figure) in which +each halo more massive than 𝑚th = 7.1 × 1010𝑀⊙ is seeded (e.g., +the Illustris project: Vogelsberger et al. 2014; Sijacki et al. 2015, +etc.); note, this seeding scheme is driven by the mass resolution of +the simulation, i.e., halos are seeded as soon as they are resolved +with a sufficient number of particles). Our model predicts that all +SMBHs formed much earlier in the universe. While a comparison +with other physical model of seeding is planned for future papers, this +figure shows the potentiality of distinguishing models by searching +for AGNs at high redshift. +We find that only a small number of mergers between seeded ha- +los occur. Table 1 shows the total number of SMBHs that formed +(𝑁SMBH,form) and the number of halos containing them at 𝑧 = 0 +(𝑁SMBH(𝑧 = 0)). Assuming efficient merging of SMBHs that are +in the same halo, then the number of mergers is Δ𝑁SMBH = +𝑁SMBH,form − 𝑁SMBH(𝑧 = 0). A feature of the Pop III.1 seeding +mechanism is that SMBHs are initially spread out from each other, +so that there are relatively few binary SMBHs and few mergers. A +detailed analysis of the mergers including the binary (and higher +order multiples) AGN number densities, and the gravitational wave +background emanating from these mergers will be discussed in a +future paper in this series. +A caveat of our seeding model is that at small redshifts, around ≲ 6, +the isolation distance in comoving units becomes so small that many +minihalos that appear after this redshift start satisfying the isolation +criteria. This effect would result in an increase in number density by +around 2 orders of magnitude by 𝑧 = 0 from the converged values +around 𝑧 ≈ 20, for all cases of 𝑑iso. However, since reionization has +completed by 𝑧 ≈ 8 (Planck Collaboration 2020), we assume that the +formation of Pop III.1 sources is also not possible below this redshift. +Hence, in our analysis, we set a limit of seed formation to be only +possible until 𝑧 = 8. For most cases of the isolation distances we +considered (≥ 75 kpc), the number density is already converged at +redshifts greater than 𝑧 = 20. However, for the case of 50 kpc, new +seeds still keep on appearing until 𝑧 = 8 (although below 𝑧 = 15 the +total number only increases by about 1%). +In Figure 3, we show a visual representation of the seeded halos in +the box at different redshifts, for all the isolation distances considered +in Fig. 2. As discussed, the 50 kpc case is the most crowded with the +highest number of seeded halos at every epoch shown. Initially all +the seeds emerge in a relatively unclustered manner, but eventually +the clustering increases as lower-mass seeded halos migrate towards +more massive halos and merge with them in overdense regions. We +perform a more detailed analysis of clustering in §3.3. +MNRAS 000, 1–13 (2023) + +SMBH formation and evolution to the local universe +5 +0 +5 +10 +15 +20 +25 +30 +35 +z +10 +4 +10 +3 +10 +2 +10 +1 +Number density [cMpc +3] +nSMBH +diso = 50 kpc +diso = 75 kpc +diso = 100 kpc +diso = 150 kpc +diso = 200 kpc +HMT model +Figure 2. The comoving number density evolution of SMBHs for different cases of the isolation distance (in proper distance). The dotted colored lines show +the total number of SMBHs, whereas the solid colored lines show the number of halos containing the black holes. The dashed green line indicates the number +density obtained from the HMT scheme, in which each halo with mass higher than 𝑚th = 7.1 × 1010𝑀⊙ is seeded (see text). The green shaded region represents +the change in number density by lowering and raising 𝑚th by a factor of 2. The black solid square indicates the estimate for the number density of SMBHs at +𝑧 = 0 by assuming each galaxy with luminosity higher than 𝐿min = 0.33𝐿∗ contains one SMBH. The black line denotes the range in 𝑛SMBH(𝑧 = 0) by varying +𝐿min from 0.1𝐿∗ to 𝐿∗. +3.2 Occupation fraction of seeded halos +From observations of local galaxies, it appears that almost all mas- +sive galaxies contain a nuclear SMBH. This implies that the SMBH +occupation fraction of halos should approach unity as halo mass +rises. Figure 4 shows the evolution of occupation fraction from one +realization of our 59.7 Mpc box, through 4 different redshifts for ha- +los ranging from [106, 1014]𝑀⊙ (the upper limit of the mass range +is chosen to include the most massive halo at 𝑧 = 0 in our 10243 +resolution simulation box, measuring 7.8 × 1013𝑀⊙). As expected, +with the decrease in the isolation distance, more and more halos are +seeded and hence the occupation fraction is higher compared to the +same mass range for larger 𝑑iso. All the fractions at 𝑧 = 0 approach +unity for the most massive halos, independent of the isolation dis- +tance. Interestingly, the most massive halo is not always occupied +by a SMBH throughout the redshift evolution in our simulations. +For example, at 𝑧 = 4 there can be significant fractions of the most +massive halos, i.e., ∼ 1012 𝑀⊙, that are not seeded. +Figure +5 +shows +the +evolution +of +the +cumulative +oc- +cupation +fraction, +i.e., +for +all +halos +more +massive +than +{108, 109, 1010, 1011, 1012, 1013}𝑀⊙, for three different cases of +isolation distance. If we consider only the most massive halos +(> 1013𝑀⊙), the fraction is close to one (as also evident from Fig. +4). At a given redshift, as we consider less massive halos, the occu- +pation fraction decreases. At a given mass threshold, as we move out +to higher redshift the occupation generally rises, since these halos +become relatively more extreme members of the global halo popula- +tion. Interestingly, the occupation fraction for all halos more massive +than 108 and 109𝑀⊙ (1010𝑀⊙ as well, although to a lower degree) +at 𝑧 = 0 differ by factors of approximately 10 among the three cases +of isolation distances considered, reflecting the same differences in +the global number densities at 𝑧 = 0 (see Fig. 2). +3.3 Clustering +We perform a clustering analysis using the corrfunc library (Sinha +& Garrison 2020) for python, and the results are shown in Fig. +6. By sampling 𝑟 in 20 logarithmic bins of 𝑟min = 0.5 Mpc/h to +𝑟max = 13.3 Mpc/h, we evaluate the 3D 2-point correlation function2 +(2pcf) 𝜉hh(𝑟) for all halos more massive than 1010𝑀⊙ at 𝑧 = 0. +Since pinocchio only evolves dark matter halos, the information of +substructures such as subhalos within halos is not stored or tracked. +This implies that only radial scales greater than the size of a typical +dark matter halo (3 to 4 Mpc at 𝑧 = 0), are relevant for consideration. +In other words, the correlation function presented here does not +include the one-halo term. From the figure, we observe that the +clustering of the SMBH-seeded halos (blue points) is always lower +compared to other cases. This is expected because of the nature of our +model, which results in larger distances between SMBHs and hence +smaller clustering amplitude. The plots for 𝑑iso = 50 and 100 kpc +clearly depict this, while the case of 200 kpc suffers from low number +statistics. The red points, which represent the clustering of random +halos with the same number and mass distribution as of the seeded +halos, are generally more than 1𝜎 higher than the blue points, except +at the largest scales. This can be clearly seen for the fiducial case of +2 All the correlation functions presented in this section have been corrected +by analytically adding large scale clustering modes corresponding to scales +larger than the box size. Refer to appendix B for more details. +MNRAS 000, 1–13 (2023) + +6 +Singh et al. +Figure 3. Projection of the positions of seeded halos (red) and non-seeded halos (blue) along the XY plane of the box for different isolation distances. The +redshift is shown in the top right corner of each panel (same for each row). Only the 30,000 most massive non-seeded halos within each panel are shown for +ease of visualisation. +MNRAS 000, 1–13 (2023) + +200 kpc +150 kpc +100 kpc +75 kpc +50 kpc +60 F +[Mpc] +30 +20 +0 +30 +50 +50 +0 +[Mpc] +Y [Mpc] +50 +0 +60 +50 +30 +20 +0 +[Mpc] +60 0 +20 +60 0 +60 0 +60 0 +20 +20 +20 +40 +60 +X [Mpc] +X [Mpc] +X [Mpc] +X [Mpc] +X [Mpc]SMBH formation and evolution to the local universe +7 +10 +6 +10 +4 +10 +2 +100 +Occupation fraction +diso = 50 kpc +diso = 100 kpc +diso = 200 kpc +108 +1010 +1012 +1014 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Occupation fraction +108 +1010 +1012 +1014 +Halo mass [M +] +108 +1010 +1012 +1014 +z = 0 +z = 4 +z = 6 +z = 10 +10 +6 +10 +4 +10 +2 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 4. Evolution of SMBH occupation fraction of halos for different cases of 𝑑iso. Top row depicts the fraction in log scale, while the bottom row shows the +same data in linear scale. The mass bins are divided into equal bins of width 0.2 dex. +0 +5 +10 +15 +20 +z +10 +4 +10 +3 +10 +2 +10 +1 +100 +Occupation fraction +diso = 50 kpc +mh > 108 +mh > 109 +mh > 1010 +mh > 1011 +mh > 1012 +mh > 1013 +0 +5 +10 +15 +20 +z +diso = 100 kpc +0 +5 +10 +15 +20 +z +diso = 200 kpc +10 +4 +10 +3 +10 +2 +10 +1 +100 +Figure 5. Cumulative occupation fractions of halos having masses greater than a given value (see legend). The shaded region represents ±1𝜎 error due to +counting statistics. +100 kpc. We also show the clustering for the fiducial case of HMT +schemes with 𝑚th = 7.1 × 1010𝑀⊙ (Sijacki et al. 2015), depicted by +green points. This model also generally shows higher clustering than +our Pop III.1 seeding model. Thus a clustering analysis of census +of a local Universe (𝑧 = 0) survey of all (or a significant fraction) +of SMBHs has the potential to distinguish between these SMBH +seeding mechanisms. +In Figure 7, we show the evolution of the projected correlation +function for the 𝑑iso =50 and 100 kpc cases (blue lines), compared to +halos with the same mass and number distribution as the respective +seeded halos (red lines). As seen in the 3D 2pcf, the clustering of the +seeded halos is always lower than the randomly selected halos and +this trend is observed even at higher redshifts. Furthermore, there +is a significant drop of the clustering amplitude of the seeded halos +for scales lower than 𝑑iso(¯𝑧form) (vertical grey band), a signature of +feedback cleared bubbles, first discussed in Paper I for 𝑧 ≥ 10. Here +we see that this signature of suppressed clustering persists to lower +redshift, although is gradually diminished as the Universe evolves to +a more clustered state. +We emphasise that comparing our clustering predictions at red- +shifts greater than 1 or 2 is not feasible with currently available obser- +vational data. The measurements from a range of luminosity of AGNs +at these redshifts imply minimum halo masses of ∼ 5 × 1011ℎ−1𝑀⊙ +at 𝑧 ∼ 3 (Allevato et al. 2014) to more than 1012ℎ−1𝑀⊙ at 𝑧 ∼ 4 (He +et al. 2018). For our 59.7 Mpc box, the number of seeded halos above +these thresholds are quite low. For instance, for the 𝑑iso =100 kpc +case, only around 6% of sources are above this threshold at 𝑧 = 3 and +only 0.7% sources are more massive than 1012ℎ−1𝑀⊙ at 𝑧 = 4. If we +apply these halo mass cuts on our seeded halos, then the clustering +signal is too noisy to make any decent comparison with the obser- +vational data. Moreover, at high halo masses the occupation fraction +approaches unity, so for the measured clustering of bright AGNs, +hosted in relatively massive halos, we expect that they may cluster +as their host halos, with no appreciable difference with respect to +MNRAS 000, 1–13 (2023) + +8 +Singh et al. +100 +101 +r [Mpc] +10 +1 +100 +hh(r) +diso = 50 kpc +Seeded halos only +All halos +HMT model +Randoms mirroring seeded +100 +101 +r [Mpc] +diso = 100 kpc +100 +101 +r [Mpc] +diso = 200 kpc +10 +1 +100 +Figure 6. The 3D 2 point correlation function for the seeded halos more massive than 1010𝑀⊙, at 𝑧 = 0 for different isolation distances. The blue points show +the correlation function for only the halos containing SMBHs, while the orange points show the correlation for all the halos, with or without a SMBH. For the red +points, we randomly select halos from the pool of all the halos, but with the same number and mass distribution as the seeded halos. The error bars indicate 1𝜎 +deviations from the mean value from randomly sampling 50 times. The green points show the correlation for halos seeded according to the halo mass threshold +(HMT) scheme, in which all the halos greater than 𝑚th = 7.1 × 1010𝑀⊙ are seeded. +rp [h +1Mpc] +5 +0 +5 +10 +15 +20 +wp(rp) [h +1Mpc] +z=1 +50 kpc +Control sample +rp [h +1Mpc] +z=3 +rp [h +1Mpc] +z=6 +rp [h +1Mpc] +z=10 +100 +101 +rp [h +1Mpc] +5 +0 +5 +10 +15 +20 +wp(rp) [h +1Mpc] +z=1 +100 kpc +Control sample +100 +101 +rp [h +1Mpc] +z=3 +100 +101 +rp [h +1Mpc] +z=6 +100 +101 +rp [h +1Mpc] +z=10 +102 +103 + [arcseconds] +102 + [arcseconds] +102 + [arcseconds] +102 + [arcseconds] +102 +103 + [arcseconds] +102 + [arcseconds] +102 + [arcseconds] +102 + [arcseconds] +Figure 7. Evolution of projected correlation function for 𝑑iso = 50 kpc (top row) and 100 kpc (bottom row) cases. The blue line is the average after computing +the correlation of the seeds from 3 orthogonal sides of the box and the shaded region represents the 1𝜎 spread. The control sample is the correlation of halos +selected randomly but with the same mass and number distribution as the seeded halos at that redshift. The red line refers to the average after randomly sampling +10 times and the shaded region refers to 1𝜎 deviations from the mean. The vertical grey line refers to the size of the isolation radius at the mean formation +redshift (𝑑iso( ¯𝑧form)) of the seeded halos, and the grey region represents 1𝜎 deviation from the mean. For 100 kpc, ¯𝑧form = 32.08, and for 50 kpc, ¯𝑧form = 27.14. +The angular axis on top of each panel corresponds to the angular scale of 𝑟𝑝 projected on the sky at the respective redshift. +MNRAS 000, 1–13 (2023) + +SMBH formation and evolution to the local universe +9 +100 +101 +rp [h +1Mpc] +101 +102 +wp(rp) [h +1Mpc] +HMT model +50 kpc +100 kpc +Zehavi et al. (2011) +Figure 8. Comparison of the results for the projected correlation function +𝑤𝑝 (𝑟𝑝) obtained from our simulations for 𝑑iso =50 kpc, 100 kpc and the +HMT scheme at 𝑧 = 0 with the observational data from Zehavi et al. (2011) +for a 𝑀𝑟 < −19.0 magnitude cut. The shaded region shows scales smaller +than the size of a typical halo at 𝑧 = 0, i.e., 𝑟𝑝 < 3ℎ−1Mpc, which are +not of interest for our comparison due to limitations of our model (lack of +sub-halos). The HMT scheme and 50 kpc models overlap, as all halos above +the threshold are seeded for that value of 𝑑iso. +currently used models. More data on AGN, especially those that are +present in lower-mass halos/galaxies is needed to test the models. +As a crude comparison, in Figure 8 we include the clustering mea- +surements from Zehavi et al. (2011), who performed the projected +clustering analysis of volume-limited sample of 570,000 galaxies +from the Seventh Data Release (Abazajian et al. 2009) of the Sloan +Digital Sky Survey (SDSS, York et al. 2000). The galaxies used in +their data extend out to 𝑧 = 0.25, with a median redshift of 𝑧 ∼ 0.1. +We compare our results at 𝑧 = 0 for 𝑑iso =50 and 100 kpc, along with +the HMT scheme, with their galaxy luminosity threshold cut result for +𝑀𝑟 < −19.0. We computed the relation between DM halo mass and +𝑟-band absolute magnitude by comparing the clustering amplitude of +pinocchio DM halos with Zehavi et al.’s measurements, minimising +the 𝜒2 of the clustering amplitude only for 𝑟 𝑝 > 3ℎ−1 Mpc (to avoid +the one-halo clustering scales); for 𝑀𝑟 < −19.0 we find a clustering- +matched halo mass of 𝑀−19.0 +PIN += 1.91 × 1012ℎ−1𝑀⊙, higher than the +value suggested in that paper (𝑀−19.0 +zehavi = 2.55 × 1011ℎ−1𝑀⊙); this +is not surprising, given the different cosmology assumed in 2011. +we then applied this halo mass cut on our 𝑑iso = 50 and 100 kpc +sources, as well as the HMT scheme, and compared the projected +correlation function for the 𝑀𝑟 < −19.0 threshold galaxies in Fig- +ure 8. For the region of interest, the clustering of the seeded halos +shows good agreement, within the errors, with the observations. The +𝑑iso = 50 kpc correlation completely overlaps the HMT one because +all the sources more massive than 𝑀−19.0 +PIN +are seeded in this model. +Also, at this high-mass cut, most of the 𝑑iso = 50 kpc sources are +also seeded in the 𝑑iso = 100 kpc model, and hence their clustering +follows similar trends. This is due to the fact that the occupation +fraction approaches unity for the most massive halos (see §3.2) for +all the isolation distances, and since the mass cut is high, this means +that most, if not all, the halos are seeded, regardless of the isolation +distance. +3.4 Ultra Deep Field +One potential way to compare our model with observational data is +to count the number of SMBHs (i.e., appearing as AGN) present +in projected deep fields of the Universe, such as the Hubble Ultra +Deep Field (HUDF). We thus create a synthetic ultra deep field +(UDF) populated with SMBHs that have formed in our simulations. +To achieve this, we use snapshots of halos at different redshifts in +the 59.7 Mpc cosmological box, using the highest resolution run. +We pierce the box orthogonally from random positions (avoiding +repetitions) and then stack the fields in redshift space to generate the +light cone of a 2.4 arcminute side length (i.e., same as the HUDF). +Figure 9 shows our constructed HUDF, for 𝑑iso = 50 kpc and 100 +kpc. The fields shown are for the redshift range 𝑧 ∈ [4, 16], with the +number of halos in the field equal to 9352 and 764 for 𝑑iso = 50 kpc +and 100 kpc, respectively. As expected, the field for the 50 kpc case +is much more densely populated with seeded halos as compared to +100 kpc. +Figure 10 shows the distribution of SMBHs within the redshift +range 𝑧 = 5 − 10 in our synthetic HUDF, where we also display the +number of sources in redshift bins of Δ𝑧 = 1. The total number of +sources in the field (last column) for the fiducial 𝑑iso =100 kpc model +is five times higher than the fiducial HMT scheme. Thus a census +of AGNs at high redshifts (𝑧 ≳ 7) can distinguish between these +models. Since the number density of sources in the HMT scheme +is quite low (effectively 0 for redshifts ≳ 8 or 9), finding even a +handful of sources at these redshifts can put stringent constrains on +this seeding scheme. In Table 2, we show the number of seeds in +the field for an extended redshift range by averaging from multiple +random realisations of the light cone, and by integrating the number +density over the field volume. Almost all the averages in the redshift +bins from the light cone are within 1𝜎 of the analytically calculated +value from the number density. The analytic numbers also show the +drastic difference in the number of sources in the different seeding +schemes at high redshifts. +4 CONCLUSIONS +We have explored the implication of the Pop III.1 seeding model for +cosmological distributions of SMBHs. This is a model that forms all +SMBHs with a single mechanism based on the change of protostellar +structure in some Pop III stars due to WIMP dark matter particle self +annihilation. This leads to reduced ionizing feedback from the pro- +tostar and efficient accretion of the baryonic content of the minihalo, +thus naturally leading to a characteristic seed mass of ∼ 105 𝑀⊙. The +model requires the Pop III.1 minihalo to form in relative isolation +from other sources. Thus the Pop III.1 seeding model involves all +SMBHs forming very early in the Universe, i.e., by 𝑧 ∼ 25, and with +a relatively unclustered initial distribution. Indeed, compared to all +other astrophysical models for SMBH formation, the Pop III.1 model +involves the earliest and least clustered distribution of seeds. This +implies that in the Pop III.1 model, black holes have plenty of time +to grow via accretion to explain the known high redshift quasars, +without the need of sustained super-Eddington accretion. +The Pop III.1 model, while being a physical model for the forma- +tion of the whole SMBH population, is relatively simple, i.e., with +only one free parameter, the isolation distance 𝑑iso. This means that +the model can be easily explored in cosmological volume simulations +that resolve minihalos, as was done first in Paper I. The constraint +of matching an estimate for the local comoving number density of +SMBHs, gives quite tight constraints on 𝑑iso ≃ 100 kpc (proper +MNRAS 000, 1–13 (2023) + +10 +Singh et al. +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +diso = 50 kpc, z +[4.00, 16.00] +4 +6 +8 +10 +12 +14 +16 +z +(a) 50 kpc +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +diso = 100 kpc, z +[4.00, 16.00] +4 +6 +8 +10 +12 +14 +16 +z +(b) 100 kpc +Figure 9. Synthetic Hubble Ultra Deep Field (HUDF) consisting of only the seeded halos for 𝑑iso = 50 kpc and 100 kpc cases over a redshift range from 4 to 16. +1 +0 +1 +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +z +[5, 6) +1159 +z +[6, 7) +1009 +z +[7, 8) +924 +z +[8, 9) +829 +z +[9, 10) +725 + diso = 50 kpc +z +[5, 10) +4646 +1 +0 +1 +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +106 +80 +84 +67 +59 + diso = 100 kpc +396 +1 +0 +1 + [arcminutes] +1.0 +0.5 +0.0 +0.5 +1.0 + [arcminutes] +52 +1 +0 +1 + [arcminutes] +19 +1 +0 +1 + [arcminutes] +3 +1 +0 +1 + [arcminutes] +0 +1 +0 +1 + [arcminutes] +0 +1 +0 +1 + [arcminutes] + mth = 7.1 × 1010M +74 +Figure 10. The distribution of SMBHs in redshift intervals in the range 𝑧 = 5 − 10 in a synthetic HUDF, where the last column shows all the sources. The first +row shows the case for 𝑑iso =50 kpc. The second row shows the case for 𝑑iso =100 kpc. The third row shows the distribution from the fiducial HMT scheme +with 𝑚th = 7.1 × 1010𝑀⊙. The total number of SMBHs in each panel are indicated in the top right corners of each. +distance). This implies most SMBHs formed at 𝑧 ≃ 30, when the +isolation distance corresponded to a comoving scale of ∼ 3 Mpc. +Following on from Paper I, we have explored the implications of the +Pop III.1 SMBH seeding model down to low redshifts, i.e., all the +way to 𝑧 = 0, which is important to allow connection to observations, +including the HUDF and local galaxy and SMBH populations. We +have also compared this model with another simple seeding scheme, +i.e., the halo mass threshold (HMT) model, that is commonly imple- +mented in cosmological volume simulations. +As presented before, all SMBHs form very early in the universe, +and their number density then remains approximately constant after +a redshift of ∼ 25. Only a small fraction of the seeded halos merge +with each other by 𝑧 = 0. The evolution of the occupation fraction +of seeded halos shows a rise to unity for the most massive halos by +𝑧 = 0. However, at intermediate redshifts there can be significant +fractions of most massive halos that are unseeded. +Our clustering analysis found that Pop III.1 seeded halos show +lower levels of clustering compared to random halos with the same +mass and number distribution as the seeded halos, at all redshifts. +However, to connect this result to observations of AGN (e.g., Allevato +et al. 2014; He et al. 2018) requires development of a SMBH growth +model, which is planned for a future paper in this series. We also +noticed a dip in the clustering of the seeded halos at scales smaller +than the isolation distance at the mean formation redshift, which +MNRAS 000, 1–13 (2023) + +SMBH formation and evolution to the local universe +11 +Table 2. Number of SMBHs in our synthetic HUDF, calculated by averaging over 100 random realizations of the light cone (From light cone column) and +by integrating the global number density (From number density column) over the redshift ranges, for 𝑑iso = 100 kpc and the fiducial HMT scheme with +𝑚th = 7.1 × 1010𝑀⊙. The errors on the averaged values correspond to 1𝜎 deviations. Note that all the numbers are rounded to the nearest integer. +z range +100 kpc +HMT +From light cone +From number density +From light cone +From number density +4-5 +110 ± 8 +101 +86 ± 19 +105 +5-6 +92 ± 6 +90 +36 ± 10 +49 +6-7 +85 ± 5 +81 +13 ± 6 +18 +7-8 +74 ± 6 +73 +3 ± 2 +7 +8-9 +69 ± 5 +66 +1 ± 1 +1 +9-10 +60 ± 5 +60 +0 +0 +10-11 +57 ± 5 +54 +0 +0 +11-12 +50 ± 5 +50 +0 +0 +12-13 +47 ± 4 +46 +0 +0 +13-14 +42 ± 4 +43 +0 +0 +14-15 +40 ± 5 +40 +0 +0 +15-16 +40 ± 5 +37 +0 +0 +is due to the feedback suppression of the isolation bubbles. This +was first discussed at 𝑧 = 10 in Paper I, and we have shown that +this suppression persists even at lower redshift, discernible down to +𝑧 ≈ 1 − 2. +To compare the clustering of our seeded halos with observational +data of galaxies, we turned to the galaxy clustering results from Ze- +havi et al. (2011). We were able to conclude that the clustering of the +seeded halos for 50 and 100 kpc isolation distances are in agreement +with the observations, after applying appropriate mass cuts on the +halo masses. The properties of binary AGN and resulting mergers, +i.e., the extreme end of the clustering signal, will be considered in +detail in a forthcoming paper in this series. +Finally, we discussed the potential of using high redshift AGN +number counts in the HUDF (or other deep fields) to differentiate +among seeding mechanisms and for constraining the value of isola- +tion distance. Detection of just a small number of SMBHs at 𝑧 ≳ 8 +would begin to discriminate between the fiducial HMT scheme and +the Pop III.1 model. +ACKNOWLEDGEMENTS +We thank Nilanjan Banik for helpful comments and useful discus- +sions. JS thanks Vieri Cammelli and Jacopo Salvalaggio for numer- +ous discussions regarding the simulations and the support of the com- +puting centre of INAF-Osservatorio Astronomico di Trieste, under +the coordination of the CHIPP project Bertocco et al. (2020); Taffoni +et al. (2020). JCT acknowledges support from ERC Advanced Grant +MSTAR. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Abazajian K. N., et al., 2009, ApJS, 182, 543 +Allevato V., et al., 2014, ApJ, 796, 4 +Banik N., Tan J. C., Monaco P., 2019, MNRAS, 483, 3592 +Barber C., Schaye J., Bower R. G., Crain R. A., Schaller M., Theuns T., 2016, +MNRAS, 460, 1147 +Begelman M. C., Volonteri M., Rees M. 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Lett., 100, 051101 +Taffoni G., Becciani U., Garilli B., Maggio G., Pasian F., Umana G., Smareglia +MNRAS 000, 1–13 (2023) + +12 +Singh et al. +Figure A1. Halo mass function of the full-resolution box at high redshift. +Lines are color-coded in redshift (see legend). Solid lines refer to pinocchio +catalogs; dashed lines to the Crocce et al. (2010) analytic fit. +R., Vitello F., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J., +Verkouter H., eds, Astronomical Society of the Pacific Conference Series +Vol. 527, Astronomical Data Analysis Software and Systems XXIX. +p. 307 (arXiv:2002.01283), doi:10.48550/arXiv.2002.01283 +Tagawa H., Haiman Z., Kocsis B., 2020, ApJ, 892, 36 +Trebitsch M., et al., 2021, A&A, 653, A154 +Tremmel M., Karcher M., Governato F., Volonteri M., Quinn T. R., Pontzen +A., Anderson L., Bellovary J., 2017, MNRAS, 470, 1121 +Vogelsberger M., et al., 2014, MNRAS, 444, 1518 +Volonteri M., Dubois Y., Pichon C., Devriendt J., 2016, MNRAS, 460, 2979 +Wang F., et al., 2021, ApJ, 907, L1 +Wise J. H., Regan J. A., O’Shea B. W., Norman M. L., Downes T. P., Xu H., +2019, Nature, 566, 85 +Yang J., et al., 2020, ApJ, 897, L14 +York D. G., et al., 2000, AJ, 120, 1579 +Zehavi I., et al., 2011, ApJ, 736, 59 +APPENDIX A: MATCHING FULL- AND +LOW-RESOLUTION PINOCCHIO RUNS +We run the 59.7 Mpc box at the full resolution of 40963 particles and +at a lower resolution of 10243 particles. These resolutions correspond +to particle masses of 1.23×105𝑀⊙ and 7.87×106𝑀⊙. The minimum +mass for halos has been set to 10 particles in both cases. Figure A1 +shows the mass function of the full-resolution box at high redshift, +where it is evident that the early growth of massive halos is slower +than in a universal model (in this case the fit to the friends-of-friends +halo mass function of Crocce et al. (2010)). We stress that there +is no reason to believe that this analytic fit is accurate at such low +masses, but we conservatively assume that the disagreement is due +to an inaccuracy of pinocchio. +Figure A2 shows the halo mass function for the low-resolution box +(thin lines) and the full-resolution run (thick lines). At high masses +the agreement of the high-resolution box with the analytic prediction +is poor, while this is not the case for the low-resolution run where the +box has not been divided into different domains. Figure A3 shows the +consistency of the seeding fraction among the high-resolution box +and a set of lower and lower resolution runs, where seeding of halos +is decided by checking which particle in Lagrangian space contains +the halos that is seeded in the full resolution box. +Figure A2. Halo mass function of the full- (thick solid lines) and low- +resolution (thin solid lines) boxes at low redshift. Lines are color-coded in +redshift (see legend). Dashed lines are the Crocce et al. (2010) analytic fit. +Figure A3. Fraction of halos of a given mass that contain a seed SMBH, +for 𝑑iso = 100 kpc. Resolution is color-coded (see legend). Thicker lines +emphasize the full-resolution (4096) and low-resolution (1024) runs. +APPENDIX B: LARGE-SCALE CLUSTERING MODES +When we use the estimators such as the corrfunc library to find +the auto correlation of halos in our 59.7 Mpc box, the correlation +function only contains the clustering modes smaller than the box +size. If we want to make a simplistic comparison of our results +with a large survey which sampled a much larger volume, we can +do so by analytically adding the larger scale clustering modes. To +understand how we achieve this, we examine the analytic expression +for calculating the 3D 2pcf for halos for the entire volume of the +Universe: +𝜉ℎℎ(𝑟) = +1 +2𝜋2 +∫ ∞ +0 +𝑑𝑘𝑘2𝑏2 +ℎ𝑃(𝑘) sin 𝑘𝑟 +𝑘𝑟 +, +(B1) +where 𝜉ℎℎ is the correlation function of halos, 𝑏ℎ is the halo bias, +and 𝑃(𝑘) is the matter power spectrum. This integral can be split in +MNRAS 000, 1–13 (2023) + +FirstBH4O.MFsathighredshift +Z=32.0000 +103 +Z=29.0000 +Z=26.0000 +Z=20.0000 +z=14.0000 +101 +Z=10.0000 +(w)uw +10-1 +10-3 +10-5 +106 +107 +108 +109 +1010 +MSeedingfractions,1ookpc,z=0 +1.0 +128 +256 +512 +0.8 +1024 +Fraction of seeded halos +4096 +0.6 +0.4 +0.2 +0.0 +9 +10 +11 +12 +13 +14 +log Mass (Mo)SMBH formation and evolution to the local universe +13 + [arcseconds] +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +w( ) +z=1 +50 kpc +Control sample + [arcseconds] +z=3 + [arcseconds] +z=6 + [arcseconds] +z=10 +102 +103 + [arcseconds] +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +w( ) +z=1 +100 kpc +Control sample +102 +103 + [arcseconds] +z=3 +102 +103 + [arcseconds] +z=6 +102 +103 + [arcseconds] +z=10 +Figure B1. Evolution of angular correlation function for 50 and 100 kpc isolation distances. The large scale modes are not added in the evaluation of this +function. The labels are the same as in Figure 7. +two parts: +𝜉ℎℎ(𝑟) = +1 +2𝜋2 +∫ 𝑘box +0 +𝑑𝑘𝑘2𝑏2 +ℎ𝑃(𝑘) sin 𝑘𝑟 +𝑘𝑟 +���������������������������������������������������������������������������� +Large scale contribution 𝜉LS(𝑟) ++ +1 +2𝜋2 +∫ ∞ +𝑘box +𝑑𝑘𝑘2𝑏2 +ℎ𝑃(𝑘) sin 𝑘𝑟 +𝑘𝑟 +���������������������������������������������������������������������� +pinocchio contribution 𝜉PIN(𝑟) += 𝜉LS(𝑟) + 𝜉PIN(𝑟), +(B2) +where 𝑘box = 2𝜋/𝐿box, with 𝐿box = 59.7 Mpc in our box. The large +scale contribution refers to the clustering modes of radial scale going +from 𝐿box to infinity, and the pinocchio contribution refers to all the +modes of radial scale from 0 to 𝐿box. Since the correlation estimator +returns 𝜉PIN, we calculated the large scale contribution by using the +linear matter power spectrum from camb python library and halo +bias from colossus python library (Diemer 2018), using the bias +model of Comparat et al. (2017), and then numerically integrated the +power spectrum to obtain 𝜉LS. +To make a direct continuation of the angular clustering as shown +in Banik et al. (2019) (their Figure 10), we present the angular clus- +tering evolution of seeded halos in figure B1 without the large scale +corrections added. This figure and Figure 7 essentially show the same +information, with the only difference that the figure presented here +is in angular scale, and without the large scale modes. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2023) + diff --git a/RNFJT4oBgHgl3EQfKSyK/content/tmp_files/load_file.txt b/RNFJT4oBgHgl3EQfKSyK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3159767c4aeb1ecc87a1b094e527a11f1749b6ba --- /dev/null +++ b/RNFJT4oBgHgl3EQfKSyK/content/tmp_files/load_file.txt @@ -0,0 +1,900 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf,len=899 +page_content='MNRAS 000, 1–13 (2023) Preprint 30 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 The formation of supermassive black holes from Population III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Evolution to the local universe Jasbir Singh,1,2,3,4★ Pierluigi Monaco,1,2,3,5 Jonathan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Tan4,6 1Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy 2INAF- Astronomical Observatory of Trieste, via Tiepolo 11, 34143 Trieste, Italy 3IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34014 Trieste, Italy 4Department of Space, Earth & Environment, Chalmers University of Technology, Gothenburg, Sweden 5INFN, Sezione di Trieste, 34149 Trieste, Italy 6Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' of Astronomy, University of Virginia, Charlottesville, VA 22904, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present predictions for cosmic evolution of populations of supermassive black holes (SMBHs) forming from Population III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', early, metal-free dark matter minihalos forming far from other sources, parameterized by isolation distance, 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Extending previous work that explored this scenario to 𝑧 = 10, we follow evolution of a (60 Mpc)3 volume to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We focus on evolution of SMBH comoving number densities, halo occupation fractions, angular clustering and 3D clustering, exploring a range of 𝑑iso constrained by observed local number densities of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We also compute synthetic projected observational fields, in particular a case comparable to the Hubble Ultra Deep Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We compare Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding to a simple halo mass threshold model, commonly adopted in cosmological simulations of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Major predictions of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 model include that all SMBHs form by 𝑧 ∼ 25, after which their comoving number densities are near-constant, with low merger rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Occupation fractions evolve to concentrate SMBHs in the most massive halos by 𝑧 = 0, but with rare cases in halos down to ∼ 108 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The 𝑑iso scale at epoch of formation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 100kpc-proper at 𝑧 ∼ 30, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', ∼ 3Mpc-comoving, is imprinted in the SMBH two-point angular correlation function, remaining discernible as a low-amplitude feature to 𝑧 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The SMBH 3D two-point correlation function at 𝑧 = 0 also shows lower amplitude compared to equivalently massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We discuss prospects for testing these predictions with observational surveys of SMBH populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Key words: black holes – formation – early universe 1 INTRODUCTION The formation of stellar mass black holes is relatively well under- stood, but the same is not true for supermassive black holes (SMBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' These black holes have masses ≥ 105𝑀⊙ and are found at the center of most large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The biggest mystery regarding their formation is explaining their high masses in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A stellar-mass BH, formed at very high redshift from the collapse of a massive primordial star, can grow by accreting gas as long as accretion can be sustained for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, this accretion is believed to be Eddington-limited by radiation pressure, so when gas inflow is abundant the growth of BH mass is expected to be exponential, with an e-fold time of ∼ 4 × 107 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Recent discoveries of high redshift quasars, for example J0313- 1806 at 𝑧 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='642 (farthest observed to date, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2021) and J1007+2115 at 𝑧 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='515 (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2020), both hosting a SMBH more massive than 109𝑀⊙ put stringent constrain on any SMBH formation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The existence of these quasars imply that these black holes grew to such high masses by the time the universe was only ∼ 700 million years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Even assuming a very early formation at 𝑧 ∼ 30, the BH seed should be at least as massive as 500 𝑀⊙ ★ E-mail: jasbir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='singh@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='it to grow to the desired mass by the observation redshift, and later formation would imply higher seed masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A variety of theories have been proposed to explain the formation of SMBHs, with different degrees of complexity and rooted to small- scale physics that is typically unresolved a cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As a consequence, simplified assumptions are typically used in these simulations to create black holes in a given dark matter halo or the galaxy contained in it, based on the properties of the parent halo or the galaxy, often using sub-grid physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' One of the simplest and widely used models is the halo mass threshold (HMT) seeding scheme based on the methods developed by Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2007) and Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2008), in which a seed black hole is assumed to form in a halo crossing a certain mass threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The Illustris project (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2014) uses this mechanism to add SMBHs of mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='4 × 105𝑀⊙ in each halo which crosses a mass threshold of 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1×1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A similar approach is used in the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulation (Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' There have been many attempts to explain the formation of SMBHs via more physical mechanisms, dating back to the last century (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Rees 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' One of the most popular mechanisms is direct collapse, which involves the collapse of a large primordial composition gas cloud in a halo of mass ∼ 108𝑀⊙ into a single supermassive star © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='11464v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='GA] 26 Jan 2023 2 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' of 104 − 106𝑀⊙ that then collapses to form a SMBH at the centre of the halo (Bromm & Loeb 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Lodato & Natarajan 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Although the number density of black holes emerging from direct collapse would be enough to explain the currently known population of high redshift quasars, the conditions required for this scenario are not thought to be common enough to explain the total observed population of SMBHs at 𝑧 = 0 (Chon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Wise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Furthermore, recent simulations have shown that the supermassive stars forming via this mechanism might not be as massive as initially predicted, but only reaching ≲ 104𝑀⊙, due to the turbulent environment present in the initial stages of galaxy formation, which disrupts the accretion flow (Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Another mechanism to form intermediate, or even supermassive black holes is through runaway stellar mergers in young and dense clusters to create massive stars as seeds of the order ∼ 200 − 103𝑀⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Portegies Zwart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This mass can be reached through repeated collisions if the massive stars can reach the cluster core to increase the collision rate drastically (Ebisuzaki 2003) before they explode as supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, predicting whether such conditions arise in galaxies and at what rate is very challenging given the the need to resolve the formation and evolution of individual stars, so pre- dictions for the cosmological population of such systems are highly uncertain (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Boekholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Chon & Omukai 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Tagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Some methods take into consideration more local properties of the host galaxy, such as the ones used in Horizon-AGN simulation (Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2016), in which the gas and stellar densities and the stellar velocity dispersion is required to exceed a certain threshold for the galaxy to be seeded with a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In addition to this, all the forming black holes must be separated by at least 50 comoving kpc, and the formation is limited until 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If all these conditions are met, the halo is seeded with a 105𝑀⊙ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Adopting a similar criteria, the more recent obelisk simulation (Trebitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2021) also applies the condition of gas and stellar density exceeding a threshold, and an isolation of 50 kpc to avoid multiple black holes forming in the same galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Furthermore, they also require the gas to be Jeans unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If all these conditions are satisfied, then a black hole of 3 × 104𝑀⊙ is assigned to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In another approach that also uses the local properties of the galaxy to assign a seed, the romulus simulation (Tremmel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2017) employs the criteria of the limit on the metallicity, a threshold on the gas density, and a temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Once all these conditions are satisfied, the mass of the seed black hole is set to be 106𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In this work, we focus on a formation scenario which invokes the Population III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 stars formed in the early universe as the progenitors of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 stars are defined to be Pop III (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', metal free) stars forming in first dark matter minihalos that are isolated from other stellar or SMBH feedback sources (McKee & Tan 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' It is assumed that in the absence of any significant radiative (or mechanical) feedback, a single dominant protostar forms at the center of the minihalo and has its structure affected by the energy input from Weakly Interacting Massive Particle (WIMP) dark matter self annihilation inside the protostar (Spolyar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Freese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Rindler-Daller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Such protostars maintain relatively cool outer layers, which allows efficient accretion of the baryonic content of the minihalo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', ∼ 105 𝑀⊙, to form a supermassive star, which subsequently collapses efficiently to a SMBH after a few Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding mechanism, which is based on locating isolated minihalos, was applied on a cosmological simulation in Banik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2019) (hereafter Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The evolution was followed from high redshifts down to 𝑧 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The main free parameter in the model is the isolation distance (𝑑iso), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', how far a newly forming minihalo needs to be from previously formed halos in order to be a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For a fiducial value of 𝑑iso = 100 kpc (proper distance), the model yields co-moving number densities of SMBHs that match the estimated level of the known 𝑧 = 0 SMBH population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Note, that in this case (and all other reasonable cases) most minihalos do not form Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Rather, most are Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 sources, which are metal free, but having been disturbed by radiative feedback undergo significant fragmentation to form only lower-mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', ∼ 10 𝑀⊙) stars (Greif & Bromm 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In this paper, we take this Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding mechanism and extend the results down to the local universe, 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In §2, we briefly describe our seeding algorithm and the tools used to apply it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Then we present our results in §3, starting with the evolution of number density of seeded halos down to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We compare these results with the HMT scheme, and also discuss the SMBH occupation fraction and cluster- ing properties of seeded halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Finally, we create synthetic Hubble Ultra Deep Fields (HUDFs) to demonstrate the possibility of using the HUDF to differentiate among different seeding mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We then present our conclusions in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2 METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 pinocchio simulations As in Paper I, to test our Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding mechanism, we used the Pinocchio code (Monaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Munari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2017) to generate a cosmological box of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc (40 ℎ−1 Mpc for ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='67) with standard Planck cosmology (Planck Collaboration 2020) and study the formation of DM (mini-)halos in that box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Pinocchio uses Lagrangian Perturbation Theory (LPT, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Moutarde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 1991) to approximate the evolution of cosmological perturbations in a ΛCDM universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For a given set of initial conditions, the code generates outputs in the form of catalogs at different redshifts, which contain mass, position and velocity of the DM halos, and a complete information of the merger histories of all the halos, with continuous time sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This code was written for applications in cosmology, where huge volumes with moderate mass resolution are requested, and its perfor- mance heavily depends on the mass resolution adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' To resolve minihalos of ∼ 106𝑀⊙ it is necessary to sample a 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box with 40963 particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' this results in a particle mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='23×105𝑀⊙, and we adopted a minimum mass of 10 particles (that would be unacceptable for an N-body simulation, but it is acceptable for a semi-analytic code like Pinocchio), resulting in a minihalo mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='23 × 106𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Such a large simulation can only be run on a super- computer, distributing the computation on a large number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since the fragmentation of collapsed particles into halos is done in Lagrangian space, the domain distributed to a task is not much larger than the dimension of the largest halo, so massive halos will not be reconstructed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As a result, with V4 of pinocchio (Munari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2017) used in Paper I, we were only able to push the simulation down to 𝑧 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We use here the novel V5 of the code, that implements a number of numerical techniques to improve memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This code will be presented elsewhere, the strategy to perform halo construction at high resolution is the following: a first step of halo construction is performed using subboxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' then the domain is augmented with all particles that lie within 𝑁Lag times the Lagrangian size of the constructed halos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' and then halo construction is performed again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) SMBH formation and evolution to the local universe 3 Memory occupation depends on 𝑁Lag, so we were forced to use 𝑁Lag = 2, while a value of 3 is a better guarantee of convergence in halo construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box with full 40963 resolution was run to z=0 on 800 MPI tasks over 100 computing nodes (each with 256 GB of RAM), so the domain was divided into 6 × 6 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 Mpc sub-volumes for halo construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The resulting halo mass function showed two problems that are presented in greater detail in an Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We discuss here their nature and their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As a consequence of the difficulty of calibrating the formation of halos with a very steep power spectrum, the mass of the first halos is underestimated by a factor of ∼ 2 at 𝑧 ∼ 30, decreasing to a negligible value at 𝑧 ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This is a known trend in pinocchio, visible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', in Figure 1 of Munari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2017) where the 𝑧 = 3 halo MF is slightly underestimated in those tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We are working to improve this prediction, but we do not consider this as a showstopper for several reasons: our seed BHs are already predicted to form very early, so this underestimation only causes us to be slightly conservative in their formation redshift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', in fact they would already have formed at slightly higher 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In our simple modeling we are assuming here immediate formation of the protostar and then the SMBH, whereas in reality this might take several Myr or even tens of Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The time span that separating 𝑧 = 32 from 𝑧 = 29 is only ∼ 14 Myr, so neglecting astrophysical timescales leads to an overestimation of formation redshift, which compensates against the underestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Finally, the minihalo threshold mass can be consider to be a second free parameter of the modeling (although one that has physical motivation to be close to 106 𝑀⊙), so one can simply consider our predictions to be valid for minihalo masses of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5×106 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We add to these arguments the fact that inaccuracies in halo masses do not propagate as inaccuracies in halo positions, that are crucial outcomes of our seeding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A more serious problem is connected to the inaccurate reconstruc- tion of halos more massive than 1012𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Indeed, the small size of the sub-box domain for constructing halos results in a poor recon- struction of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This problems makes predictions at 𝑧 = 0 unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We thus produced the same box at a lower resolution, sampled with 10243 particles, on a single MPI task on a 256 GB node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Again, this was possible thanks to V5 of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In this case halo construction is as good as it can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, the identification of halos that contain seed SMBHs has been performed in the high resolution box, and though the simulations share the same large-scale structure, matching massive halos in the two boxes is not a clean pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We then resorted to this algorithm: starting from the fact that one low-resolution particle contains 64 high-resolution ones, we cal- culated which particle in the lower resolution box includes the seeded mini-halo, and assigned the seed to the halo that contains that specific low-resolution particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We checked that results at 𝑧 = 0 produced with the low- and high-resolution simulations were consistent, with a significant difference in halo clustering of halos more massive than a certain threshold that is an expected consequence of the inaccurate mass reconstruction and the known relation of halo bias with halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In the following we will present results at 𝑧 = 0 based on the low resolution box, unless mentioned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 Seeding scheme To determine which halos are seeded with a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star and thence SMBH, consider the scenario depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 1, unfolding in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The figure shows three stars A, B and C in different halos where only A and C become Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 stars whereas B is a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 star, depending on the separation and formation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Star A Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A schematic illustration of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 SMBH seeding scenario depicting the conditions for a star to be isolated enough to be considered as a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' formed first, which then influenced its environment within a sphere of radius equal to 𝑑feedback, expected to be primarily radiative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since this star is in a pristine primordial gas without the influence of any feedback from nearby stars, it is defined to be a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Star B, which subsequently forms at a distance less than 𝑑feedback from star A, is affected by the feedback and hence is a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 star (or even a Pop II star if it has been chemically polluted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Finally, star C forms outside the sphere of influence of both A and B, and is thus also assigned to be a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star and thus a SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For the model considered here, the feedback distance is set equal to the isolation distance 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' So effectively, the condition for a star to be regarded as a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star is that when it is forming, there should be no previously formed halos present in the sphere of radius 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We consider 𝑑iso as a free parameter in our theory and vary it to match the observed number density of the SMBHs in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='3 Seed identification in the dark matter catalogs To perform the seed identification analysis from the dark matter cata- logs generated by pinocchio, we first divided the entire redshift range (from 𝑧 = 0 to the redshift when the first minihalo forms, 𝑧 ≈ 40) into small bins of widths ranging from Δ𝑧 = 1, 2 or 3, depending on the output catalogs available, which in turn depends on the relative change in positions of (mini)halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The bins are wider at high red- shifts, but smaller at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Then for each redshift interval (𝑧𝑙, 𝑧ℎ] where (𝑧ℎ > 𝑧𝑙), we utilised k-d tree data structure to create a three dimensional map in position space of all the halos existing be- tween 𝑧ℎ and 𝑧𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The positions used to create the tree are taken from the output catalog of pinocchio at the lower redshift of the interval (𝑧𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since the positions are not updated once the tree is constructed, we account for the change in the positions within this redshift interval by finding the maximum change (𝛿) of position among all the halos existing for the entire redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Then for each minihalo crossing the mass threshold of 106𝑀⊙ (or as in the nomenclature of pinoc- chio: "appearing") at a redshift 𝑧app ∈ (𝑧𝑙, 𝑧ℎ], we perform a ball search using the k-d tree to find all the halos around the appearing minihalo within a sphere of radius 𝑑iso − 2𝛿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If there exists even a single halo at the redshift 𝑧app within this sphere, then this minihalo 1 A factor of 2 is multiplied with 𝛿 to account for the change in position of both the minihalo at the center of the sphere and all the other halos within the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) A: Pop Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 Afeedback = ★ B: Pop Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 C: Pop Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='14 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' is flagged as a halo containing a non-Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 star at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If there are no halos existing at this redshift, then the ball search is performed again with the same minihalo at the center, but this time within a sphere of radius 𝑑iso + 2𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Then for all the halos existing at redshift 𝑧app within the shell of radius 𝑑iso ± 2𝛿, we find the exact distance between the minihalo at the center and all these halos using the exact positions at 𝑧app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If this distance is greater than 𝑑iso for all the halos within the shell, then the minihalo at the center is flagged as a Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 source, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', an SMBH-seeded halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This process is repeated for each minihalo crossing the threshold mass within the two redshifts, and then this whole procedure is performed again for all the redshift intervals, until the whole redshift range is covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In this way we are able check the isolation condition for each minihalo appearing in the cosmological box and find all the seeded minihalos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' At smaller redshifts, the change in positions of the halos (𝛿) within the redshift intervals becomes comparable to the isolation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This implies that the quantity 𝑑iso − 2𝛿 can become negative (in our simulation box, this happens at around 𝑧 ≈ 15 for 𝑑iso = 50 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In this case, the ball search is directly performed in a sphere of radius 𝑑iso + 2𝛿, and then the exact distances between the minihalo at the center and all the other halos existing at 𝑧app is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This division of the entire redshift interval and creating the k-d only at specific redshifts is performed to avoid reconstructing the tree with the up-to-date position at every instance a new minihalo appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since the number of minihalos is very large, it becomes highly expensive computationally to reconstruct the tree with updated positions each time a new minihalo appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 Number density evolution As explained in the last section and in detail in Paper I, we identify SMBH-seeded halos by the condition that the isolation sphere of radius 𝑑iso around a newly forming minihalo is not be populated by any other existing halo (of mass greater than our minihalo threshold mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The obtained results for the evolution of number density for different values of 𝑑iso (in proper distance units) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The estimate for the observed number density of SMBHs in the local Universe, 𝑛SMBH(𝑧 = 0) (black square in the figure) is calculated by assuming that each galaxy with luminosity greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='33𝐿∗ hosts a SMBH (see Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Here 𝐿∗ is the characteristic luminosity corresponding to 𝑀B = −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 + 5 log ℎ = −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='55 (for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Norberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The colored dotted lines show the number density evolu- tion of total number of SMBHs, whereas the colored solid lines show the number density for seeded halos (which can be slightly smaller due to mergers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' These results are from the highest resolution sim- ulation with 40963 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Compared to the number densities in Figure 1 of Paper I, the values obtained here are slightly lower (by a factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='45 for 100 kpc, and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='65 for 50 kpc) because we have considered periodic boundary conditions when identifying the seeds, which was not done in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2, it can be clearly seen that as the isolation distance is reduced, the number of formed SMBHs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This is expected because smaller 𝑑iso results in more halos satisfying the isolation criteria for hosting SMBH seeds within our simulation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We can also conclude that for a certain range of 𝑑iso (≈ 90 kpc to 170 kpc), the number density obtained is in reasonable agreement with the 𝑧 = 0 estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A key feature of the fiducial Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 SMBH seeding model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', with 𝑑iso = 100 kpc, is that all SMBHs have formed very early in the Universe: the process is essentially complete by 𝑧 ≃ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Total number of formed SMBHs (𝑁SMBH,form), total number of SMBHs remaining at 𝑧 = 0 assuming efficient mergers (𝑁SMBH(𝑧 = 0)), the difference between these (Δ𝑁SMBH = 𝑁SMBH,form − 𝑁SMBH(𝑧 = 0)), which is equivalent to the number of mergers, and the fraction of original SMBHs that are destroyed by mergers ( 𝑓merger = Δ𝑁SMBH/𝑁SMBH,form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 𝑑iso [kpc] 𝑁SMBH,form 𝑁SMBH(𝑧 = 0) Δ𝑁SMBH 𝑓merger 50 15470 14499 971 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='28 75 3394 3303 91 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='68 100 1234 1222 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='97 150 306 306 0 0 200 121 121 0 0 We compare this prediction to a halo mass threshold model (HMT scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' shown by the green dashed line in the figure) in which each halo more massive than 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙ is seeded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', the Illustris project: Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2015, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' note, this seeding scheme is driven by the mass resolution of the simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', halos are seeded as soon as they are resolved with a sufficient number of particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Our model predicts that all SMBHs formed much earlier in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' While a comparison with other physical model of seeding is planned for future papers, this figure shows the potentiality of distinguishing models by searching for AGNs at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We find that only a small number of mergers between seeded ha- los occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Table 1 shows the total number of SMBHs that formed (𝑁SMBH,form) and the number of halos containing them at 𝑧 = 0 (𝑁SMBH(𝑧 = 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Assuming efficient merging of SMBHs that are in the same halo, then the number of mergers is Δ𝑁SMBH = 𝑁SMBH,form − 𝑁SMBH(𝑧 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A feature of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding mechanism is that SMBHs are initially spread out from each other, so that there are relatively few binary SMBHs and few mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A detailed analysis of the mergers including the binary (and higher order multiples) AGN number densities, and the gravitational wave background emanating from these mergers will be discussed in a future paper in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' A caveat of our seeding model is that at small redshifts, around ≲ 6, the isolation distance in comoving units becomes so small that many minihalos that appear after this redshift start satisfying the isolation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This effect would result in an increase in number density by around 2 orders of magnitude by 𝑧 = 0 from the converged values around 𝑧 ≈ 20, for all cases of 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, since reionization has completed by 𝑧 ≈ 8 (Planck Collaboration 2020), we assume that the formation of Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 sources is also not possible below this redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Hence, in our analysis, we set a limit of seed formation to be only possible until 𝑧 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For most cases of the isolation distances we considered (≥ 75 kpc), the number density is already converged at redshifts greater than 𝑧 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, for the case of 50 kpc, new seeds still keep on appearing until 𝑧 = 8 (although below 𝑧 = 15 the total number only increases by about 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In Figure 3, we show a visual representation of the seeded halos in the box at different redshifts, for all the isolation distances considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As discussed, the 50 kpc case is the most crowded with the highest number of seeded halos at every epoch shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Initially all the seeds emerge in a relatively unclustered manner, but eventually the clustering increases as lower-mass seeded halos migrate towards more massive halos and merge with them in overdense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We perform a more detailed analysis of clustering in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) SMBH formation and evolution to the local universe 5 0 5 10 15 20 25 30 35 z 10 4 10 3 10 2 10 1 Number density [cMpc 3] nSMBH diso = 50 kpc diso = 75 kpc diso = 100 kpc diso = 150 kpc diso = 200 kpc HMT model Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The comoving number density evolution of SMBHs for different cases of the isolation distance (in proper distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The dotted colored lines show the total number of SMBHs, whereas the solid colored lines show the number of halos containing the black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The dashed green line indicates the number density obtained from the HMT scheme, in which each halo with mass higher than 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙ is seeded (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The green shaded region represents the change in number density by lowering and raising 𝑚th by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The black solid square indicates the estimate for the number density of SMBHs at 𝑧 = 0 by assuming each galaxy with luminosity higher than 𝐿min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='33𝐿∗ contains one SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The black line denotes the range in 𝑛SMBH(𝑧 = 0) by varying 𝐿min from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1𝐿∗ to 𝐿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 Occupation fraction of seeded halos From observations of local galaxies, it appears that almost all mas- sive galaxies contain a nuclear SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This implies that the SMBH occupation fraction of halos should approach unity as halo mass rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure 4 shows the evolution of occupation fraction from one realization of our 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box, through 4 different redshifts for ha- los ranging from [106, 1014]𝑀⊙ (the upper limit of the mass range is chosen to include the most massive halo at 𝑧 = 0 in our 10243 resolution simulation box, measuring 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='8 × 1013𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As expected, with the decrease in the isolation distance, more and more halos are seeded and hence the occupation fraction is higher compared to the same mass range for larger 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' All the fractions at 𝑧 = 0 approach unity for the most massive halos, independent of the isolation dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Interestingly, the most massive halo is not always occupied by a SMBH throughout the redshift evolution in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For example, at 𝑧 = 4 there can be significant fractions of the most massive halos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', ∼ 1012 𝑀⊙, that are not seeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure 5 shows the evolution of the cumulative oc- cupation fraction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', for all halos more massive than {108, 109, 1010, 1011, 1012, 1013}𝑀⊙, for three different cases of isolation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If we consider only the most massive halos (> 1013𝑀⊙), the fraction is close to one (as also evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' At a given redshift, as we consider less massive halos, the occu- pation fraction decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' At a given mass threshold, as we move out to higher redshift the occupation generally rises, since these halos become relatively more extreme members of the global halo popula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Interestingly, the occupation fraction for all halos more massive than 108 and 109𝑀⊙ (1010𝑀⊙ as well, although to a lower degree) at 𝑧 = 0 differ by factors of approximately 10 among the three cases of isolation distances considered, reflecting the same differences in the global number densities at 𝑧 = 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='3 Clustering We perform a clustering analysis using the corrfunc library (Sinha & Garrison 2020) for python, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' By sampling 𝑟 in 20 logarithmic bins of 𝑟min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 Mpc/h to 𝑟max = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='3 Mpc/h, we evaluate the 3D 2-point correlation function2 (2pcf) 𝜉hh(𝑟) for all halos more massive than 1010𝑀⊙ at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since pinocchio only evolves dark matter halos, the information of substructures such as subhalos within halos is not stored or tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This implies that only radial scales greater than the size of a typical dark matter halo (3 to 4 Mpc at 𝑧 = 0), are relevant for consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In other words, the correlation function presented here does not include the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' From the figure, we observe that the clustering of the SMBH-seeded halos (blue points) is always lower compared to other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This is expected because of the nature of our model, which results in larger distances between SMBHs and hence smaller clustering amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The plots for 𝑑iso = 50 and 100 kpc clearly depict this, while the case of 200 kpc suffers from low number statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The red points, which represent the clustering of random halos with the same number and mass distribution as of the seeded halos, are generally more than 1𝜎 higher than the blue points, except at the largest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This can be clearly seen for the fiducial case of 2 All the correlation functions presented in this section have been corrected by analytically adding large scale clustering modes corresponding to scales larger than the box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Refer to appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 6 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Projection of the positions of seeded halos (red) and non-seeded halos (blue) along the XY plane of the box for different isolation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The redshift is shown in the top right corner of each panel (same for each row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Only the 30,000 most massive non-seeded halos within each panel are shown for ease of visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 200 kpc 150 kpc 100 kpc 75 kpc 50 kpc 60 F [Mpc] 30 20 0 30 50 50 0 [Mpc] Y [Mpc] 50 0 60 50 30 20 0 [Mpc] 60 0 20 60 0 60 0 60 0 20 20 20 40 60 X [Mpc] X [Mpc] X [Mpc] X [Mpc] X [Mpc]SMBH formation and evolution to the local universe 7 10 6 10 4 10 2 100 Occupation fraction diso = 50 kpc diso = 100 kpc diso = 200 kpc 108 1010 1012 1014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 Occupation fraction 108 1010 1012 1014 Halo mass [M ] 108 1010 1012 1014 z = 0 z = 4 z = 6 z = 10 10 6 10 4 10 2 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Evolution of SMBH occupation fraction of halos for different cases of 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Top row depicts the fraction in log scale, while the bottom row shows the same data in linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The mass bins are divided into equal bins of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 0 5 10 15 20 z 10 4 10 3 10 2 10 1 100 Occupation fraction diso = 50 kpc mh > 108 mh > 109 mh > 1010 mh > 1011 mh > 1012 mh > 1013 0 5 10 15 20 z diso = 100 kpc 0 5 10 15 20 z diso = 200 kpc 10 4 10 3 10 2 10 1 100 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Cumulative occupation fractions of halos having masses greater than a given value (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The shaded region represents ±1𝜎 error due to counting statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We also show the clustering for the fiducial case of HMT schemes with 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙ (Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2015), depicted by green points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This model also generally shows higher clustering than our Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Thus a clustering analysis of census of a local Universe (𝑧 = 0) survey of all (or a significant fraction) of SMBHs has the potential to distinguish between these SMBH seeding mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In Figure 7, we show the evolution of the projected correlation function for the 𝑑iso =50 and 100 kpc cases (blue lines), compared to halos with the same mass and number distribution as the respective seeded halos (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As seen in the 3D 2pcf, the clustering of the seeded halos is always lower than the randomly selected halos and this trend is observed even at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Furthermore, there is a significant drop of the clustering amplitude of the seeded halos for scales lower than 𝑑iso(¯𝑧form) (vertical grey band), a signature of feedback cleared bubbles, first discussed in Paper I for 𝑧 ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Here we see that this signature of suppressed clustering persists to lower redshift, although is gradually diminished as the Universe evolves to a more clustered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We emphasise that comparing our clustering predictions at red- shifts greater than 1 or 2 is not feasible with currently available obser- vational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The measurements from a range of luminosity of AGNs at these redshifts imply minimum halo masses of ∼ 5 × 1011ℎ−1𝑀⊙ at 𝑧 ∼ 3 (Allevato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2014) to more than 1012ℎ−1𝑀⊙ at 𝑧 ∼ 4 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For our 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box, the number of seeded halos above these thresholds are quite low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For instance, for the 𝑑iso =100 kpc case, only around 6% of sources are above this threshold at 𝑧 = 3 and only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7% sources are more massive than 1012ℎ−1𝑀⊙ at 𝑧 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If we apply these halo mass cuts on our seeded halos, then the clustering signal is too noisy to make any decent comparison with the obser- vational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Moreover, at high halo masses the occupation fraction approaches unity, so for the measured clustering of bright AGNs, hosted in relatively massive halos, we expect that they may cluster as their host halos, with no appreciable difference with respect to MNRAS 000, 1–13 (2023) 8 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 100 101 r [Mpc] 10 1 100 hh(r) diso = 50 kpc Seeded halos only All halos HMT model Randoms mirroring seeded 100 101 r [Mpc] diso = 100 kpc 100 101 r [Mpc] diso = 200 kpc 10 1 100 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The 3D 2 point correlation function for the seeded halos more massive than 1010𝑀⊙, at 𝑧 = 0 for different isolation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The blue points show the correlation function for only the halos containing SMBHs, while the orange points show the correlation for all the halos, with or without a SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For the red points, we randomly select halos from the pool of all the halos, but with the same number and mass distribution as the seeded halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The error bars indicate 1𝜎 deviations from the mean value from randomly sampling 50 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The green points show the correlation for halos seeded according to the halo mass threshold (HMT) scheme, in which all the halos greater than 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙ are seeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' rp [h 1Mpc] 5 0 5 10 15 20 wp(rp) [h 1Mpc] z=1 50 kpc Control sample rp [h 1Mpc] z=3 rp [h 1Mpc] z=6 rp [h 1Mpc] z=10 100 101 rp [h 1Mpc] 5 0 5 10 15 20 wp(rp) [h 1Mpc] z=1 100 kpc Control sample 100 101 rp [h 1Mpc] z=3 100 101 rp [h 1Mpc] z=6 100 101 rp [h 1Mpc] z=10 102 103 [arcseconds] 102 [arcseconds] 102 [arcseconds] 102 [arcseconds] 102 103 [arcseconds] 102 [arcseconds] 102 [arcseconds] 102 [arcseconds] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Evolution of projected correlation function for 𝑑iso = 50 kpc (top row) and 100 kpc (bottom row) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The blue line is the average after computing the correlation of the seeds from 3 orthogonal sides of the box and the shaded region represents the 1𝜎 spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The control sample is the correlation of halos selected randomly but with the same mass and number distribution as the seeded halos at that redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The red line refers to the average after randomly sampling 10 times and the shaded region refers to 1𝜎 deviations from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The vertical grey line refers to the size of the isolation radius at the mean formation redshift (𝑑iso( ¯𝑧form)) of the seeded halos, and the grey region represents 1𝜎 deviation from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For 100 kpc, ¯𝑧form = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='08, and for 50 kpc, ¯𝑧form = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The angular axis on top of each panel corresponds to the angular scale of 𝑟𝑝 projected on the sky at the respective redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) SMBH formation and evolution to the local universe 9 100 101 rp [h 1Mpc] 101 102 wp(rp) [h 1Mpc] HMT model 50 kpc 100 kpc Zehavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2011) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Comparison of the results for the projected correlation function 𝑤𝑝 (𝑟𝑝) obtained from our simulations for 𝑑iso =50 kpc, 100 kpc and the HMT scheme at 𝑧 = 0 with the observational data from Zehavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2011) for a 𝑀𝑟 < −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 magnitude cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The shaded region shows scales smaller than the size of a typical halo at 𝑧 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 𝑟𝑝 < 3ℎ−1Mpc, which are not of interest for our comparison due to limitations of our model (lack of sub-halos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The HMT scheme and 50 kpc models overlap, as all halos above the threshold are seeded for that value of 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' currently used models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' More data on AGN, especially those that are present in lower-mass halos/galaxies is needed to test the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As a crude comparison, in Figure 8 we include the clustering mea- surements from Zehavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2011), who performed the projected clustering analysis of volume-limited sample of 570,000 galaxies from the Seventh Data Release (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2009) of the Sloan Digital Sky Survey (SDSS, York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The galaxies used in their data extend out to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='25, with a median redshift of 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We compare our results at 𝑧 = 0 for 𝑑iso =50 and 100 kpc, along with the HMT scheme, with their galaxy luminosity threshold cut result for 𝑀𝑟 < −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We computed the relation between DM halo mass and 𝑟-band absolute magnitude by comparing the clustering amplitude of pinocchio DM halos with Zehavi et al.’s measurements, minimising the 𝜒2 of the clustering amplitude only for 𝑟 𝑝 > 3ℎ−1 Mpc (to avoid the one-halo clustering scales);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' for 𝑀𝑟 < −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 we find a clustering- matched halo mass of 𝑀−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 PIN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='91 × 1012ℎ−1𝑀⊙, higher than the value suggested in that paper (𝑀−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 zehavi = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='55 × 1011ℎ−1𝑀⊙);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' this is not surprising, given the different cosmology assumed in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' we then applied this halo mass cut on our 𝑑iso = 50 and 100 kpc sources, as well as the HMT scheme, and compared the projected correlation function for the 𝑀𝑟 < −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 threshold galaxies in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' For the region of interest, the clustering of the seeded halos shows good agreement, within the errors, with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The 𝑑iso = 50 kpc correlation completely overlaps the HMT one because all the sources more massive than 𝑀−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 PIN are seeded in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Also, at this high-mass cut, most of the 𝑑iso = 50 kpc sources are also seeded in the 𝑑iso = 100 kpc model, and hence their clustering follows similar trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This is due to the fact that the occupation fraction approaches unity for the most massive halos (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='2) for all the isolation distances, and since the mass cut is high, this means that most, if not all, the halos are seeded, regardless of the isolation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='4 Ultra Deep Field One potential way to compare our model with observational data is to count the number of SMBHs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', appearing as AGN) present in projected deep fields of the Universe, such as the Hubble Ultra Deep Field (HUDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We thus create a synthetic ultra deep field (UDF) populated with SMBHs that have formed in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' To achieve this, we use snapshots of halos at different redshifts in the 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc cosmological box, using the highest resolution run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We pierce the box orthogonally from random positions (avoiding repetitions) and then stack the fields in redshift space to generate the light cone of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='4 arcminute side length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', same as the HUDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure 9 shows our constructed HUDF, for 𝑑iso = 50 kpc and 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The fields shown are for the redshift range 𝑧 ∈ [4, 16], with the number of halos in the field equal to 9352 and 764 for 𝑑iso = 50 kpc and 100 kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As expected, the field for the 50 kpc case is much more densely populated with seeded halos as compared to 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure 10 shows the distribution of SMBHs within the redshift range 𝑧 = 5 − 10 in our synthetic HUDF, where we also display the number of sources in redshift bins of Δ𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The total number of sources in the field (last column) for the fiducial 𝑑iso =100 kpc model is five times higher than the fiducial HMT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Thus a census of AGNs at high redshifts (𝑧 ≳ 7) can distinguish between these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since the number density of sources in the HMT scheme is quite low (effectively 0 for redshifts ≳ 8 or 9), finding even a handful of sources at these redshifts can put stringent constrains on this seeding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' In Table 2, we show the number of seeds in the field for an extended redshift range by averaging from multiple random realisations of the light cone, and by integrating the number density over the field volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Almost all the averages in the redshift bins from the light cone are within 1𝜎 of the analytically calculated value from the number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The analytic numbers also show the drastic difference in the number of sources in the different seeding schemes at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 4 CONCLUSIONS We have explored the implication of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding model for cosmological distributions of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This is a model that forms all SMBHs with a single mechanism based on the change of protostellar structure in some Pop III stars due to WIMP dark matter particle self annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This leads to reduced ionizing feedback from the pro- tostar and efficient accretion of the baryonic content of the minihalo, thus naturally leading to a characteristic seed mass of ∼ 105 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The model requires the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 minihalo to form in relative isolation from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Thus the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeding model involves all SMBHs forming very early in the Universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', by 𝑧 ∼ 25, and with a relatively unclustered initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Indeed, compared to all other astrophysical models for SMBH formation, the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 model involves the earliest and least clustered distribution of seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This implies that in the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 model, black holes have plenty of time to grow via accretion to explain the known high redshift quasars, without the need of sustained super-Eddington accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 model, while being a physical model for the forma- tion of the whole SMBH population, is relatively simple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', with only one free parameter, the isolation distance 𝑑iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This means that the model can be easily explored in cosmological volume simulations that resolve minihalos, as was done first in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The constraint of matching an estimate for the local comoving number density of SMBHs, gives quite tight constraints on 𝑑iso ≃ 100 kpc (proper MNRAS 000, 1–13 (2023) 10 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] diso = 50 kpc, z [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='00, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='00] 4 6 8 10 12 14 16 z (a) 50 kpc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] diso = 100 kpc, z [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='00, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='00] 4 6 8 10 12 14 16 z (b) 100 kpc Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Synthetic Hubble Ultra Deep Field (HUDF) consisting of only the seeded halos for 𝑑iso = 50 kpc and 100 kpc cases over a redshift range from 4 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 1 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] z [5, 6) 1159 z [6, 7) 1009 z [7, 8) 924 z [8, 9) 829 z [9, 10) 725 diso = 50 kpc z [5, 10) 4646 1 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] 106 80 84 67 59 diso = 100 kpc 396 1 0 1 [arcminutes] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0 [arcminutes] 52 1 0 1 [arcminutes] 19 1 0 1 [arcminutes] 3 1 0 1 [arcminutes] 0 1 0 1 [arcminutes] 0 1 0 1 [arcminutes] mth = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010M 74 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The distribution of SMBHs in redshift intervals in the range 𝑧 = 5 − 10 in a synthetic HUDF, where the last column shows all the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The first row shows the case for 𝑑iso =50 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The second row shows the case for 𝑑iso =100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The third row shows the distribution from the fiducial HMT scheme with 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The total number of SMBHs in each panel are indicated in the top right corners of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This implies most SMBHs formed at 𝑧 ≃ 30, when the isolation distance corresponded to a comoving scale of ∼ 3 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Following on from Paper I, we have explored the implications of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 SMBH seeding model down to low redshifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', all the way to 𝑧 = 0, which is important to allow connection to observations, including the HUDF and local galaxy and SMBH populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We have also compared this model with another simple seeding scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', the halo mass threshold (HMT) model, that is commonly imple- mented in cosmological volume simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' As presented before, all SMBHs form very early in the universe, and their number density then remains approximately constant after a redshift of ∼ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Only a small fraction of the seeded halos merge with each other by 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The evolution of the occupation fraction of seeded halos shows a rise to unity for the most massive halos by 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, at intermediate redshifts there can be significant fractions of most massive halos that are unseeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Our clustering analysis found that Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 seeded halos show lower levels of clustering compared to random halos with the same mass and number distribution as the seeded halos, at all redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' However, to connect this result to observations of AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', Allevato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' 2018) requires development of a SMBH growth model, which is planned for a future paper in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We also noticed a dip in the clustering of the seeded halos at scales smaller than the isolation distance at the mean formation redshift, which MNRAS 000, 1–13 (2023) SMBH formation and evolution to the local universe 11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Number of SMBHs in our synthetic HUDF, calculated by averaging over 100 random realizations of the light cone (From light cone column) and by integrating the global number density (From number density column) over the redshift ranges, for 𝑑iso = 100 kpc and the fiducial HMT scheme with 𝑚th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 × 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The errors on the averaged values correspond to 1𝜎 deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Note that all the numbers are rounded to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' z range 100 kpc HMT From light cone From number density From light cone From number density 4-5 110 ± 8 101 86 ± 19 105 5-6 92 ± 6 90 36 ± 10 49 6-7 85 ± 5 81 13 ± 6 18 7-8 74 ± 6 73 3 ± 2 7 8-9 69 ± 5 66 1 ± 1 1 9-10 60 ± 5 60 0 0 10-11 57 ± 5 54 0 0 11-12 50 ± 5 50 0 0 12-13 47 ± 4 46 0 0 13-14 42 ± 4 43 0 0 14-15 40 ± 5 40 0 0 15-16 40 ± 5 37 0 0 is due to the feedback suppression of the isolation bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This was first discussed at 𝑧 = 10 in Paper I, and we have shown that this suppression persists even at lower redshift, discernible down to 𝑧 ≈ 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' To compare the clustering of our seeded halos with observational data of galaxies, we turned to the galaxy clustering results from Ze- havi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We were able to conclude that the clustering of the seeded halos for 50 and 100 kpc isolation distances are in agreement with the observations, after applying appropriate mass cuts on the halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The properties of binary AGN and resulting mergers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', the extreme end of the clustering signal, will be considered in detail in a forthcoming paper in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Finally, we discussed the potential of using high redshift AGN number counts in the HUDF (or other deep fields) to differentiate among seeding mechanisms and for constraining the value of isola- tion distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Detection of just a small number of SMBHs at 𝑧 ≳ 8 would begin to discriminate between the fiducial HMT scheme and the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Nilanjan Banik for helpful comments and useful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' JS thanks Vieri Cammelli and Jacopo Salvalaggio for numer- ous discussions regarding the simulations and the support of the com- puting centre of INAF-Osservatorio Astronomico di Trieste, under the coordination of the CHIPP project Bertocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Taffoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' JCT acknowledges support from ERC Advanced Grant MSTAR.' 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Xu H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 2019, Nature, 566, 85 Yang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 2020, ApJ, 897, L14 York D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 2000, AJ, 120, 1579 Zehavi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=', 2011, ApJ, 736, 59 APPENDIX A: MATCHING FULL- AND LOW-RESOLUTION PINOCCHIO RUNS We run the 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box at the full resolution of 40963 particles and at a lower resolution of 10243 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' These resolutions correspond to particle masses of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='23×105𝑀⊙ and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='87×106𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The minimum mass for halos has been set to 10 particles in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure A1 shows the mass function of the full-resolution box at high redshift, where it is evident that the early growth of massive halos is slower than in a universal model (in this case the fit to the friends-of-friends halo mass function of Crocce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' We stress that there is no reason to believe that this analytic fit is accurate at such low masses, but we conservatively assume that the disagreement is due to an inaccuracy of pinocchio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure A2 shows the halo mass function for the low-resolution box (thin lines) and the full-resolution run (thick lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' At high masses the agreement of the high-resolution box with the analytic prediction is poor, while this is not the case for the low-resolution run where the box has not been divided into different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure A3 shows the consistency of the seeding fraction among the high-resolution box and a set of lower and lower resolution runs, where seeding of halos is decided by checking which particle in Lagrangian space contains the halos that is seeded in the full resolution box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Halo mass function of the full- (thick solid lines) and low- resolution (thin solid lines) boxes at low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Lines are color-coded in redshift (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Dashed lines are the Crocce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2010) analytic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Fraction of halos of a given mass that contain a seed SMBH, for 𝑑iso = 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Resolution is color-coded (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Thicker lines emphasize the full-resolution (4096) and low-resolution (1024) runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' APPENDIX B: LARGE-SCALE CLUSTERING MODES When we use the estimators such as the corrfunc library to find the auto correlation of halos in our 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc box, the correlation function only contains the clustering modes smaller than the box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' If we want to make a simplistic comparison of our results with a large survey which sampled a much larger volume, we can do so by analytically adding the larger scale clustering modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' To understand how we achieve this, we examine the analytic expression for calculating the 3D 2pcf for halos for the entire volume of the Universe: 𝜉ℎℎ(𝑟) = 1 2𝜋2 ∫ ∞ 0 𝑑𝑘𝑘2𝑏2 ℎ𝑃(𝑘) sin 𝑘𝑟 𝑘𝑟 , (B1) where 𝜉ℎℎ is the correlation function of halos, 𝑏ℎ is the halo bias, and 𝑃(𝑘) is the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This integral can be split in MNRAS 000, 1–13 (2023) FirstBH4O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='MFsathighredshift Z=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0000 103 Z=29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='0000 Z=26.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Evolution of angular correlation function for 50 and 100 kpc isolation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The large scale modes are not added in the evaluation of this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The labels are the same as in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' two parts: 𝜉ℎℎ(𝑟) = 1 2𝜋2 ∫ 𝑘box 0 𝑑𝑘𝑘2𝑏2 ℎ𝑃(𝑘) sin 𝑘𝑟 𝑘𝑟 ���������������������������������������������������������������������������� Large scale contribution 𝜉LS(𝑟) + 1 2𝜋2 ∫ ∞ 𝑘box 𝑑𝑘𝑘2𝑏2 ℎ𝑃(𝑘) sin 𝑘𝑟 𝑘𝑟 ���������������������������������������������������������������������� pinocchio contribution 𝜉PIN(𝑟) = 𝜉LS(𝑟) + 𝜉PIN(𝑟), (B2) where 𝑘box = 2𝜋/𝐿box, with 𝐿box = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content='7 Mpc in our box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' The large scale contribution refers to the clustering modes of radial scale going from 𝐿box to infinity, and the pinocchio contribution refers to all the modes of radial scale from 0 to 𝐿box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' Since the correlation estimator returns 𝜉PIN, we calculated the large scale contribution by using the linear matter power spectrum from camb python library and halo bias from colossus python library (Diemer 2018), using the bias model of Comparat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2017), and then numerically integrated the power spectrum to obtain 𝜉LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' To make a direct continuation of the angular clustering as shown in Banik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' (2019) (their Figure 10), we present the angular clus- tering evolution of seeded halos in figure B1 without the large scale corrections added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This figure and Figure 7 essentially show the same information, with the only difference that the figure presented here is in angular scale, and without the large scale modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQfKSyK/content/2301.11464v1.pdf'} diff --git a/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/2301.00853v1.pdf.txt b/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/2301.00853v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9055d613160d6d98a572df8621654ec78a0f42c1 --- /dev/null +++ b/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/2301.00853v1.pdf.txt @@ -0,0 +1,1218 @@ +(2022), 1–17 +R&D INTERNSHIP REPORT +Tweets’ popularity dynamics +Ferdinand Willemin +Sahar [1], France +Email: ferdinand.willemin@ecl20.ec-lyon.fr. +28th August 2022 +Abstract +This article charts the work of a 4 month project aimed at automatically identifying patterns of tweets’ popularity evolution using Machine +Learning and Deep Learning techniques. To apprehend both the data and the extent of the problem, a straightforward clustering algorithm +based on a point to point distance is used. Then, in an attempt to refine the algorithm, various analyses especially using feature extraction +techniques are conducted. Although the algorithm eventually fails to automate such a task, this exercise raises a complex but necessary issue +touching on the impact of virality on social networks. +Keywords: IA, TIME-SERIES, TWITTER, VIRALITY, CLUSTERING, HDBSCAN, BIG DATA, TV DENOISING +1. +INTRODUCTION +1.1 +Context and stakes +Sahar is a private company specialized in collecting, processing +and visualizing massive open-access data available in the web, +including social networks data. The firm provides a data anal- +ysis tool tailored to very different types of clients, requiring it +to be both exhaustive and flexible. +This work aims to comprehend popularity mechanisms +within social networks in an attempt to help improve some +of Sahar’s tool features. +Nowadays, Twitter is the ultimate medium for informa- +tion broadcasting. It encompasses more than 200 millions +usersa around the world and is likely to serve both individuals +as well as institutions. Moreover, its recent effects over politics +and the economy have placed it at the core of global attention. +This is why we picked it to study popularity mechanisms on +social networks. +1.2 +Twitter +Twitter [2] is a social network, and more specifically a micro- +blogging service, where users can post and interact with +messages known as tweets. A user can interact with a tweet +in three different ways : he can comment it, retweet it or like +it. To comment a tweet is to publicly answer a tweet, with +a message that will appear in the tweet’s comment section. To +retweet means to display a tweet on one’s own profile, in order +to share it with one’s followers (people that are aware of your +activity on the network). To like is to show one’s consensus +and/or support to a message. Tweets are limited to 280 char- +acters which makes them as much easily broadcastable as +highly ephemeral. Tweets often come with hashtags : key- +words preceded by the typographic sign # added with the aim +ahttps://www.blogdumoderateur.com/chiffres-twitter/ +of specifying themes the tweet resonates to. +Figure 1. Screen shot of a tweet and its interaction buttons. [3] +The manner in which Tweets spread across the network +— i.e. their way to be shared or seen by users — is atypical. +That is why a study dedicated to them is conducted. +1.3 +Definition of the problem +The problem can be formulated as follows : Can a reasonable +number of interpretable patterns showing the dynamics of tweets’ +popoularity through time be identified ? +In order to provide a sufficient answer to the above prob- +lematic, certain subjective terms must be defined and precised. +The exploratory (vs. being solely results-driven) nature of this +work has opened to door to the following precisions: the pop- +ularity is arbitrarily designated by the number of likes, a rea- +arXiv:2301.00853v1 [cs.LG] 2 Jan 2023 + +CliffSchecter +@cliffschecter +As a reminder we have zero proofZEROgun laws +work.Youknow,if you don't include Japan,U.K. +Belgium,Canada,Iceland,Romania,Norway,Austria +Argentina,Netherlands,SouthKorea,Italy,Greece, +Chile,France,Spain,Sweden,Singapore,Portugal, +Israel,Czechia,Denmark, +1:38AM·May25,2022+TwitterforiPhone +36.8KRetweets +1,184QuoteTweets +197.4K Likes +Don +7 +C +[Like +Tweet your reply +Reply2 +Tweets’ popularity dynamics +sonable number is likely to be below 20 and interpretable +patterns must be distinct enough so that we can name and +describe them unambiguously. +Note : In this work, we treat popularity and virality to be the +strictly identical. +1.4 +State of the art +Research on this topic emanates from the rise of social net- +works which is a relatively recent phenomenon. The year +2008 can be considered the representative year of this ten- +dency, as it constitutes Facebook’s passing over 100 million +users [4], making it the first social network at the time. Twitter +reached this count in 2011 [5]. +The first noticeable study dealing with popularity dynam- +ics in user-generated content [6] is applied to the video host +YouTube, with the aim of finding locally relevant content dif- +fering from the well-known most popular content that only +considers mass appeal and an instantaneous vision. The analysis +leads to the identification of three categories : the junk, the +quality and the viral video dynamics. Junk videos undergo +a burst of popularity which drops quickly afterwards because +they do not spread through the social network. Quality videos +meet a very sudden peak in popularity, certainly caused by +an exogenous effect (such as being featured on the first page +of YouTube), and a subsequently passive decline. Viral videos +face a slowly increase up to a peak followed by a slow de- +crease which reflects a word-of-mouth process. Concurrently, +most of the videos do not experience any peak in popularity, +embodying a fourth category : the silent videos. +YouTube is actually not a social network but some parallels +can be drawn between its features and Twitter’s. For instance, +the social network displays, alike Youtube, the trendiest topics +on its first page which can trigger exogenous effects as well. +However, apart from the difference of contents’ nature be- +tween the two websites, the article [6] is based on applying a +mathematical model (the self-excited Hawkes conditional Poisson +process) to the data. The assumptions made to use it are not +detailed enough and may not fit with our case. As a result, the +origins of the four categories are not fully defined. +Another interesting work [7] implements a clustering al- +gorithm to gather hashtags’ popularity dynamics with similar +shapes. The popularity is measured by the number of appear- +ances of a hashtag over time. In summary, it uses a K-Means +algorithm provided by an "improved" euclidean distance that +ignores both the overall hashtags’ appearances and the temporal +gap between the popularity major peaks. +Although the idea of using a clustering algorithm may +seem appropriate — since such algorithms are used to classify +objects according to specified features — some of the choices +made by the author hide some conjectures that are arguable +in the context of our work. First, given two tweets and the +history of their number of likes, does having the same shape +but at different scales with a time lag can be deemed to be +similar dynamics ? Figure 2 illustrates this issue by showing +two evolutions whose shapes could be considered as similar +as they both include successively a quick growth, an effective +stage, a slower but longer rise and eventually a cap. However, +their maximum number of likes (nmax +likes) vary with a factor of +almost 30 ! The processes that generated them do not engage +the same range thus their dynamics are different. The use of +a K-Means algorithm raises other questions, especially around +the choice of the K hyperparameter which sets the number of +clusters. Moreover, the preprocessing adopted —implying to +manipulate the 1000 most frequently mentioned hashtags — +does not guarantee to lead to a large enough dataset where all +patterns possible are represented. +Figure 2. Two tweet’s history of likes designated by their Twitter id. +Ahmed et al. [8] also makes use of a clustering strategy. +They build a two-dimensional feature space and a correlation- +based similarity metric combined with the affinity propagation +algorithm [9] to extract the main evolution patterns from their +datasets. Their approach is worth exploring but its complexity +— especially the simultaneous analysis of two distant measures +carrying information of different nature — makes it a method +for further investigations that should be implemented at a later +stage. Unfortunately, time did not allow our study to experi- +ment with this model. +A different way of seeing things, presented by C. Li, J. +Liu, and S. Ouyang [10], represents the evolution of videos’ +popularity from the chinese service provider Youku with a +succession of two states : 1 if the video is experiencing a burst of +popularity and 0 otherwise, the popularity corresponding to the +number of views in a day. A burst stands for a striking increase +in popularity. Quantitatively, it is represented by an exceed- +ing of a certain threshold by the derivative, whose value +is actually established at three times the average derivative. +The idea of extracting key information from curves to discrim- +inate them is encouraging but requires additional work to be + +3 +carried out in order to gain in rigor. +Among others, this representation may only embrace +sketchy tendencies and by doing so neglect some nuances. +For instance, integrating the size or duration of the different +states could allow us to distinguish subgroups between them +or might reveal some typical behaviors related to them. More +specifically, the threshold’s value must come with a reliable +justification that exhibit a true social phenomenon and the +burst’s mathematical definition itself may need to be re- +viewed to improve its robustness. +All in all, many lines of attack have been exposed over +the previous years. They should now be used to meticulously +define a tailor-made strategy which will attempt to answer the +initial problem. +1.5 +Organization of the article +Once the data presented (2), we will implement a HDB- +SCAN clustering algorithm [11] specifically designed for +our study (3). Then, in an attempt to produce customizable +and better results, another version of the HDBSCAN algo- +rithm will be applied on a transformed dataset (4). By doing so, +an original method created to reduce and store information +of the popularity’s history into what is called a tweet vector will +be introduced. Ultimately, an overall assessment of the tech- +niques developed followed by a list of promising avenues for +further exploration of the topic will be proposed (5). +2. +PRESENTATION OF THE DATA +2.1 +The data fetching system +Part of Sahar’s solution is an intelligent web scrapingb feature +of Twitter. It notably enables users to analyze all messages +containing one or several keywords — commonly a hashtag — +and all the ones published by one or several given users. Thus, +collecting an important chunk of messages around a particular +subject is made extremely efficient and smooth. +There are two main advantages to using a topic-oriented +scraping. On the one hand, tracking all tweets posted in a +given period of time would be too costly both in time and +energy : around 500 millions tweets are produced each day +[12] and assuming that their average lifetimec is a few days, each +dataset would hold billions of tweets. On the other hand, it +is conceivable that evolutionary patterns will vary depend- +ing on the topics addressed. Hence, Sahar’s tool’s filtering +capability prevent us from mixing the topics — and therefore +the patterns — too much. +Furthermore, let us recall that the project’s conditions (1.1) +demand the virality algorithm to be compatible with the +other features of Sahar’s tool, as they are to work in synergy. +bAct of collecting publicly available data from the web, often on a large +scale. +cDuration starting when the message is published and ending when it no +longer generates interactions. +From all the above, it only seems natural that we use the com- +pany’s own scrapping method. +To ensure a significant diversity regarding the amplitude +of the data monitored, the keywords or accounts followed were +selected for their capacity to generate many likes. With- +out this precision, we would automatically collect all tweets, +especially noisy ones that do not generate any interaction. In +effect, these inert tweets constitute the large majority of the +publications [13]. Besides, by removing those inert tweets not +only do we save time, but we also prevent our datasets from +being excessively large. +2.2 +The resulting datasets +Two datasets are used in this study. The first one, named +dataset 1, is composed of 2785 tweets observed from 23/05/2022 +to 27/05/2022 included. The points of each time series are +recorded every 10 minutes. The second one, dataset 2, is +composed of 3000 tweets monitored from 17/06/2022 to +26/06/2022 included with a record every 5 minutes. The +keywords and users associated to each one of them are detailed +in Appendix 1. +N.B: It must be kept in mind that the frequency of data ac- +quisition is not always constant in practice, due to the web +scrapping feature performances. +2.3 +Rudimentary preprocessing +Figure 3 shows the evolution of popularity right after it had +been collected. The exact time is not specified to not overload +the plot although it has a 1-second accuracy. +Figure 3. Typical raw data from the 1st dataset. +This raw data format must undergo a first and obvious pre- +processing (Figure 4). To enable comparison between the +different evolutions, the x axis is expressed in terms of duration +in SI units. In the meantime, a large part of the asymptotic +behavior is removed since it doesn’t carry any relevant infor- +mation. To do so, we store the values t10 and t95 (respectively +the times at which the curve reach 10% and 95% of its maxi- +mum value). This draws an interval ∆t = t95 – t10. Then, we +"extend" the interval by 20% by defining tmax = t10 + 1, 2.∆t. +Finally, the data from 0 to tmax provides a window focused on +the evolution dynamics. + +4 +Tweets’ popularity dynamics +Figure 4. Example of the raw data and its elementary values (on top) trans- +formed into a more suitable format (on the bottom). +3. +CLUSTERING +Clustering belongs to the unsupervised class of machine learn- +ing algorithm because it doesn’t involve two steps including +a training stage where inputs are entered simultaneously as +their corresponding outputs. On the contrary, it engages a +single step where inputs are agglomerated to form clusters +according to their distance to each other. Therefore, building +a metric which represents how close the popularity evo- +lutions are is fundamental. Because it truely influences the +quality of the results, it is the touchy phase of the process. +To design such a model, it is essential to answer the follow- +ing questions beforehand : what does having similar popularity +dynamics means ? and what is a good cluster in our case ? +Through a naive approach, we answer the first problem by +looking for a point to point proximity. The second issue is +more tricky since it implies subjective criteria. For instance, +one may prefer to get homogeneous clusters regarding the +number of tweets it contains without being demanding on the +groups’ inner proximity, while another may want to obtain +very accurate clusters, whatever the quantity of curves they +hold. To get around such complications, quality is arbitrarily +favored, taking into account that our notion of proximity is +by nature very demanding. +In addition, the algorithm chosen in this study comes with +a useful feature : the existence of a noise cluster, where all the +unique dynamics — i.e. those that are "far" to every other — +are placed. The size of this extra lot is also a criteria that can +lead to new compromises. +3.1 +Choice of the metric +A prior search of the existing work conducted to the testing +of two different distances : the Dynamic Time Warping distance +and what we call the L1 distance. The first one naturally allows +to avoid the main problem caused by the inaccurate recording +of the data (cf 2.2) : the disparity of the values along the +time axis. The L1 distance doesn’t possess the same ability +so a more elaborated preprocessing is needed to tackle this +issue. However, the latter is faster to compute and more +respectful of the temporal distribution. +3.1.1 +Dynamic Time Warping distance +The Dynamic Time Warping distance (or DTW distance) +between two time series A and B is introduced in [14]. It is +determined as follow : +Let {a1, . . . , ap} and {b1, . . . , bq} respectively be the values of +A and B along the Y axis (note that p and q don’t have to be +equal). Then, let W designate the set of all the paths between +A and B, i.e. the selections of (i, j) where every points from +A and B are involved at least once. Let also consider that a +simple distance δ, typically the euclidean distance, has been +settled. For all i ∈ {1, . . . , p} and j ∈ {1, . . . , q}, δ(ai, bj) is +calculated. The results can be visualized in a distance matrix +D = +� +di,j +� +1≤i≤p +1≤j≤q +where di,j = δ(ai, bj) (figure 5). The DTW +distance is then defined as +DTW(A, B) = min +w∈W +� +� � +(i,j)∈w +δ(ai, bj) +� +� +(1) +The path w that lead to the DTW distance is called the warping +path (see figures 5 and 7). +This is equivalent to applying a non-linear temporal distortion +to the data which aligns on the time axis points that are the +closest towards the Y axis and then return the sum of their +distances. +Figure 5. Visualization of a fictional distance matrix and its warping path (in +red). +Many aspects of this definition are physically incoherent +: for a given i, calculating δ(ai, bj) ∀j means that the tempo- +ral dimension is neglected. To address these obstacles, some +correcting constraints are added : +• the boundary constraint stipulates that a path must include (1, +1) and (p, q) which guarantees to consider a beginning +and end notion ; +• the monotonicity constraint states that ik ≤ ik+1 ∀k and +jk ≤ jk+1 ∀k which compels the points inspection to al- +ways advance in time ; + +~ 100% +95% +10%D = +di, +S(a1, b1 +s(ai, b1) +a15 +• the continuity constraint imposes that |ik+1 – ik| ≤ 1 ∀k and +|jk+1–jk| ≤ 1∀k which forces comparing distances between +neighbor points, and therefore to progress in both time +series "at the same speed" ; +• the warping window specifies a maximum range (R) of +points to visit to prevent the points scan to be "stuck" in +one time series while it keep going in the other : it can be +expressed as ∀k, |ik – jk| ≤ R. +Figure 6. Illustration of the DTW correcting constraints and their efect on +the algorithm : the monotonicity constraint in red, the continuity constraint +ingreenandthewarpingwindowinpurple. Here,thepathcanonlycontinue +to the green locations that are not in hatched zones. +Once these restrictions are employed, a recursive formula +can be revealed to find a suitable warping path : +γ(i, j) = δ(i, j) + min[γ(i – 1, j), γ(i – 1, j – 1), γ(i, j – 1)] +(2) +where γ(i, j) stands for the cumulative distance until point (i, j). +[14] also proposes an algorithm to determine this distance +— the DTW algorithm — which runs in quadratic time (O(pq)). +However, an optimization of the correcting constraints com- +bined with an appropriate reduction of the data (called Ab- +straction approach) can result in an upgrade that works in linear +time : the FastDTW [15]. The latter is used for our calcula- +tions. +3.1.2 +"L1" Distance +A more straightforward method consists in computing a point +to point distance between the curves. However, comparing +them at similar instants requires to have their points aligned +along the time axis and to carry the same amount of points, +otherwise the distance cannot be determined. As mentioned +before (section 3.1), our data doesn’t satisfy these conditions. +To overpass this issue, the curves must be interpolated +linearly with a fixed step [16]. To limit the loss of informa- +tion caused by this process the step is set at 5min, i.e. twice +as small as our recording frequency. To ensure that they +Figure 7. Illustration of the warping path (black lines) between two tweets. +all possess the same number of points, they are extended by +considering they have reached their asymptote : points +equal to the last existing one are added until they get to the +longest evolution.d +Conclusively, given n ∈ N∗ and two interpolated time +series A and B with their respective points {a1, . . . , an} and +{b1, . . . , bn}, the distance is expressed by : +dL1 = +n +� +i=1 +|ai – bi| +(3) +In its continuous form (i.e. when step → 0), (3) is written +dL1 = +tmax +� +0 +|f – g|dx where f and g are the functions describing +the two evolutions, which inspired the name L1 distance. It +coincides with the surface visible between the two curves +(figure 8). The lower this area is, the closer the dynamics are. +Figure 8. Illustration of the L1 distance between two tweets. +dNote : With the benefit of hindsight, it would have been easier and +more legitimate to modify the rudimentary preprocessing (section 2.3) by +truncating the asymptotes at the maximum tmax of the dataset. + +R +D= +.+ +di+1,i+1 +a1,6 +di +a1Msg n°1529195078606106625 +Msg n°1529612363883810819Tweet n°1529612363883810819 +Tweet n°15291950786061066256 +Tweets’ popularity dynamics +3.1.3 +Analysis of the distances +Discriminating the metrics isn’t trivial. Indeed, their main goal +is to bring the equivalent curves closer together and to move +away from each other those that are different, so they must be +judged on this task. However, they were built for this very +purpose. The fact that they serve as both a test object and +an evaluation instrument forces us to imagine alternative +criteria of performance. +Knowing that the human eyes are the best tools to execute +this task for a low number of tweets, a qualitative survey is +first conducted. For each distance the ith closest pair of curves +is examined and compared (i taking different values). Besides, +some test samples are selected. Each one contains two pairs +of tweets : one considered as close and the other as a distant pair +according to human eyes. The goal is to see whether the pairs +are labeled identically with our metrics. To make it happen, +a random sub-sampling of the 1st dataset is implemented +which selects 200 tweets out of 3278. This is necessary be- +cause the DTW algorithm is quite time consuming. As a +reference, it takes about 5 seconds to calculate the pairwise L1 +distance between the sub-sampled data (including the inter- +polation step) while it takes about 30 minutes with the DTW +distance. Incidentally, this facet counts as a great prejudice +against the latter. +This inquiry reveals some weaknesses of DTW, as it some- +times leads to doubtful results (figure 9). On the contrary, the +L1 distance does not experience such outcomes. +Even if a qualitative examination is not sufficient to con- +clude, it exhibits another proof of DTW’s inferiority. As +shown in figure 10, this distance is slightly influenced by the +number of records present in the time series. Despite some +attempts to get rid of this drawback (mainly through a penal- +ization strategy), it remains disturbing. Bypassing the problem +with preprocessing would be absurd since it constitutes the +main strength of the algorithm. +All in all, due to its better robustness, simplicity and +speed, the L1 distance is chosen as the reference distance for +the rest of the project. However, DTW is worth exploring. +Its inherent upsides may be useful for other applications and +can be greatly improved in other contexts. As for the compu- +tational time, one must notice that the algorithms used may +not be fully optimized. +3.1.4 +Adding a weighting +Afterwards, it has been assumed that the beginning of the +popularity evolution was carrying more information about +the underlying social dynamics than the end. To introduce +this feature into our metric, a weighting is added, favoring +the first instants and handicapping the last ones. +Concretely, it implies multiplying the L1 distance with a +penalty function. Several functions were studied to eventu- +ally select the one that offered the most control. It is defined +as : +f (t) = (1 – ϵ) ∗ +�th(β – α.t) + 1 +2 +� ++ ϵ +(4) +where : +• β is such that f (0) = 0, 99 ; +• α is such as f (tmax) = 0.7 with tmax being the median of all +the tmax of the dataset ; +• ϵ = 0.05 stipulating that the minimum weight is 5% +The weighting’s influence can be seized in figure 11. +Figure 11. Graph showing the impact of the weighting on the L1 distance. +Note : the amplitude of the weighting has been increased for the figure. +3.2 +The clustering algorithm +3.2.1 +HDBSCAN +HDBSCAN [17] is originally a DBSCAN [18] upgrade. Both +are density-based algorithm, which means they have been +conceived to identify dense areas within groups of points. As +mentioned before, one of their considerable advantages is their +capacity to generate a noise cluster where all the unique dynam- +ics are stored. Actually, all time series that are not in a dense +zone are considered as noise. +To execute such a task, it needs to be provided with two +main parameters : min_samples and min_cluster_size. +min_samples is used to define a disk surrounding each +data point whose ray corresponds to the distance between a +point and its min_samples’s closest neighbor. Two points +whose L2 distance is inferior to both point’s rays will be con- +sidered to be in the same dense area. If only one of the two +belongs to the disk of the other, their distance will be increased +to match with the bigger ray. Therefore, min_samples is +used to redefine the space topology by accentuating den- +sity phenomena: the points which are not in dense areas are +even more isolated. The distance created by this process is +called the mutual reachability distance. +min_cluster_size corresponds to the minimum size +for a dense area to be considered as a cluster. Bellow this +value the time series inside the zone are viewed as noise. To + +Tweetn°1529595180587929600 +Tweetn°1529969772011610144 +Weightingfunction7 +Figure 9. Two pairs of tweets extracted and their "proximity rank". According to the DTW distance, the lef pair contains closer tweets than in the right one. +Qualitatively, one can judge the opposite. +Figure 10. Variation of the average DTW distance according to the number of points (raw plot on the lef, without extreme points on the right). It seems that +time series carrying a lot of information (i.e. many records) have a tendency to move away from the others relatively to the DTW distance. +better seize its impact, a deeper comprehension of the algo- +rithm is needed. +Let’s consider the minimum spanning tree of the dataset +[19] built from the mutual reachability distance. By defini- +tion, this tree links all the points together each time with a +single bound. Let’s consider that when a point is linked with +another, they both form a cluster. Let’s delete the bounds one +by one from the biggest to the smallest. Removing a bound is +equivalent to divide a parent cluster into two children clusters. +Therefore at the end, all the points are isolated from the others. +All the divisions triggered during the process can be visualized +in a dendogram as in figure 12. +Since this method leads to an enormous amount of clus- +ters, a refinement is added : rather than seeing each split as a +parent cluster giving birth to two children clusters, it could +be interpreted as a "cluster erosion" : if a children is too small +to be a cluster itself, it means that the parent cluster didn’t +actually split, but instead lost some points that became noise. +Hence, min_cluster_size is there to define what too small +means. Figure 13 shows the kind of change provoked on the +dendogram. +Figure12. Dendogramdepictingclusters’splitsasϵ(herebeingnormalized +and called distance) decreases. [11] + +Correlationbetweennumberofpointsand DTWdistance +1e6 +6 +/ distance to the othercurves +5 +3 +0 +0 +500 +1000 +1500 +2000 +2500 +Amount of records withinthe time seriesCorrelationbetweennumberofpointsand DTWdistance +32000 +curves +31900 +other +31800 +to the +31700 +/distance +31600 +31500 +DTW +31400 +Average +31300 +31200 +0 +500 +1000 +1500 +2000 +2500 +Amountof recordswithinthetimeseries0.9 +6.4 +0.8 +5.6 +0.7 +4.8 +0.6 +log(Number of points) +4.0 +distance +0.5 +3.2 +0.4 +2.4 +0.3 +1.6 +0.2 +0.1 +0.8 +0.0 +0.08 +Tweets’ popularity dynamics +Figure 13. +The same dendogram afer applying min_cluster_size +� +λ = 1 +ϵ +� +[11] +Eventually, a flat clustering is extracted by keeping the most +stable groups that don’t overlap with each others. Crudely, +they relate to the longest ones in the dendogram. +This stands for the basic knowledge necessary to under- +stand our way of using HDBSCAN. A more exhaustive and +detailed explanation can be found in the library documentation +[11]. +3.2.2 +Parameters selection +Building a score to choose the parameters in a way that guar- +antees both a reasonable number of clusters and a sufficient +proximity within them was first intended. Unfortunately, it +didn’t succeed mainly because it was too sensitive : the score +obtained after an important variation of the parameters — +around 10 units — was so close that the "optimal" parameters +could dramatically change between the datasets. These param- +eters are so affecting that such a behavior cannot be allowed. +As a result, another indicator is adopted : the size of +the noise cluster. Usually, this quantity is relatively large +— from 40 to 50% of the dataset — which is quite annoy- +ing. Hence, it is desirable to reduce it as much as possible. +The best lowering is achieved with the lowest values of both +min_cluster_size and min_samples, i.e. 2. It is not sur- +prising at all knowing their influence on the clustering (see +3.2.1). Considering that the accuracy of the clusters is fa- +vored over a reasonable amount of groups (cf 3), setting both +min_samples=2 and min_cluster_size=2 is tolerable. Ap- +plying these values on the merged datasets (5786 tweets) +triggers 45,37% noise (2625 tweets). +3.2.3 +Iterative clustering +Even when the parameters are adjusted to minimize the noise +rate, it remains really high (cf 3.2.2). Although knowing which +time series have unique dynamics is important, such noise rate +is excessive and may come from disproportionate proximity +requirements. To overcome that, an iterative clustering is pro- +posed. Basically, each iteration consists in considering the +noise cluster as a dataset itself on which the HDBSCAN +algorithm is applied (figure 14). At each round the density +notion is redefined since the number of inputs is lower, so new +clusters emerge. As they are of poorer quality, the round in +which they appeared is stored to be reminded when clusters +are shown. +Figure 14. Scheme representing the iterative clustering principle. +The iteration stops when the noise share is bellow 5% of +the entire dataset. +3.3 +Results +In Appendix 2, the 18 first clusters obtained from each dataset +are displayed (figure 20, 21 and 22). Table 1 summarizes in- +formation about the clusters obtained. +Table 1. Summary of the clustering results +Dataset +1 +2 +Both +Number of tweets +2785 +3001 +5786 +Number of clusters +491 +498 +975 +Noise rate +1,1% +1,0% +1,4% +Average cluster size +6 tweets +6 tweets +6 tweets +Standard +deviation +of the clusters size +9 tweets +14 tweets +17 tweets +Highest cluster size +141 tweets +232 tweets +349 tweets +Undoubtedly, these outcomes are not sufficient to answer +our original problem (section 1.3). The reasonable number of +clusters is far from being reached and some of them are too +disparate to be clearly describable (see figure 15). On top of +that, clusters made of popular tweets — those with high values +of nmax +likes — are often too small to be considered as pattern repre- +sentatives, although this is to be expected given our parameters +selection (section 3.2.2). +Several causes could explain such poor results : + +0 +100 +90 +1 +80 +2 +70 +3 +Number of points +60 +lue +va +4 +50 +40 +5 +30 +9 +20 +7 +10 +8 +0Noise cluster +Dataset +Clusters +Clustering +Noise cluster +Dataset +Clusters +Clustering9 +Figure 15. Biggest cluster from the 1st dataset. +• the study might suffer from a lack of data : datasets might +be too little for each pattern to be represented by an ap- +propriate number of tweets ; +• discovering a reasonable amount of interpretable patterns +may not be possible because of the plurality of social +dynamics ; +• our definition of evolution dynamics itself (section 3) +may be too extreme or at least not accurate : so far, two +curves have the exact same dynamics if they are equal at any +point (without considering the weighting). +In the remainder of this article, a revision of this last +point is proposed, as it seems to be the most obvious source of +error. +4. +THE TWEET VECTOR +Instead of measuring how close popularity evolutions are at +any point, the study will now focus on the essential partic- +ularities of what we call dynamics. Then a clustering can be +exclusively applied on those attributes. As it corresponds to a +relaxation of our previous distance measure, it may lead to a +lower amount of clusters without compromising their quality. +Every element that can characterize a time series’ dynamics +will be stored in a vector called the tweet vector. +4.1 +Components of the tweet vector +The dynamics’ characteristics identified are : +• nmax +likes — the maximum number of like — as it reflects the +intensity of the dynamics ; +• slopemean — the average slope of the curve — since it de- +scribes how fast the dynamics are ; +• the key instants, i.e. the times at which the evolution reaches +a given percentage of its maximum value, to take into +account its temporal distribution ; +• the boosts’ raw increases (see 4.1.2) which constitutes a +striking distinction between evolutions ; +• a compressed format of the L1 distance between the current +curve and the others, which carries indications about its +overall (and relative) shape. +Although nmax +likes and slopemean are quite explicit values, the +integration of the key instants, boosts and L1 distance com- +ponents require some additional work. In doing so, we will +attempt to obtain a tweet vector as small as possible, since the +idea is to only carry necessary information. +4.1.1 +The key instants +It is assumed that extracting tp with p ∈ {10, 20, 30, 40, 50, 60, +70, 80, 90} (as manipulated is section 4) is more than enough +to apprehend the temporal distribution of the time series. The +objective is to select the most relevant ones too reduce the +tweet vector’s size. To tackle this task, the well-known Prin- +cipal Component Analysis (PCA) algorithm is used. Details +about its theoretical functioning can be found in [20]. +The PCA’s implementation of scikit-learn [21] allows to +be aware the share of variance kept during the dimension +reduction process. It also offers the possibility to visualize a +projection of the variance distribution of the different key +instants over a given dataset. +After running it on the dataset, it turns out that retaining +the 3 first components allows to keep 95% of the variance +and that the less correlated triplet of key instants is (t10, t50 +and t90). Although PCA’s components are not equals to the +key instants themselves, we can suppose this particular triplet +contains the information we need. +4.1.2 +The boosts +A boost designates a sudden change of rhythm favoring +an increase of popularitye (see figure 16). It is a specific +behavior of the virality evolutions which could become an +efficient tool to describe a given pattern. +Figure 16. Qualitative identification of boosts in a given popularity evolu- +tion. +Instinctively, the mathematical definition must integrate a +condition concerning the derivative of the evolution. How- +ever, data imperfections can make the task complicated (figure +17). +eboosts are inspired by the bursts described in [10] + +Tweet n°1529236803160461312 +6000 +5000 +4000 +boosts +3000 +slowdowns +2000 +1000 +end of evolution +0 +0 +20000 +40000 +60000 +80000 +100000 +120000 +140000 +Time (s)10 +Tweets’ popularity dynamics +Figure 17. Raw derivative of the time series window : the second boost does +not stand out enough to be identified. +A consistent way to highlight boosts while smoothing the +derivative is to employ Total Variation Denoising (TV De- +noising). This technique consists in approaching the deriva- +tive with a piecewise constant curve. This is equivalent to +solving the following problem : +find Rmin = argmin (f (O, .)) +where f (O, R) = ∥O – R∥2 + λ. ∥∇R∥1 +Here, O is the original derivative values, R is the vector repre- +senting the reconstructed derivative and λ is a scalar affecting +the "penalty" applied to the variation ∇R : the higher λ is, +the less variations the reconstructed curve contains. In prac- +tice, this parameter is chosen so that the maximum number +of boosts in the dataset doesn’t exceed 6. Beyond 6, some +boosts are likely to be mistakes caused by an excessive sensitiv- +ity. +Thanks to the cvxpy python library [22], this method is +easily implementable. +Boosts correspond to the denoised derivative’s bumps. +In order to detect them properly, an encoding program is +performed. First, the variations of the denoised derivative are +computed. After removing the numerical noise, the signs of it +are extracted. Hence, each tweet is related to a vector made of +1, -1 and 0. Finally, the boosts are enumerated and identified +knowing that they start with a 1 and end as soon as a -1 is en- +countered. Moreover, if the first (resp. the last) sign is -1 (resp. +1), it means the curve begins (resp. ends) with a boost. Figure +18 illustrates the different steps of the process and spotlights +the resulting boosts. +The raw increase of boosts is added to the tweet vector. +Hence, 6 components should be dedicated to it with the ith +component representing the ith boost of the evolution (or equal +to 0 if there is no ith boost). However, the 3 first boosts seem +to be the most determining : a tweet having 4 boosts is +often almost inert, so that a single like is considered as a great +increase. That is why only three components are eventually +allocated to this characteristic. +4.1.3 +The L1 distance integration +To be integrated to the tweet vector, the L1 distance matrix +— containing all the pairwise L1 distances of the dataset — +is reduced with an auto-encoder neural network. With- +out detailing the fundamentals of deep learning, this model +is built with two symmetrical networks : the encoder and the +decoder. The first one is trained to encrypt the data in a lower +dimensional space while the second one is trained to recon- +struct the initial data based on the encryption (also called latent +space). If the decoder succeeds in retrieving the original data, it +means the latent space carries the essential information of +the matrix. The architecture of our model for the 2nd dataset +is shown in figure 19 : for each tweet, 4 components are +enough to describe its distance to all the others. The ELU +activation function [23] and smooth L1 loss function [24] are +used. +Figure 19. Scheme representing the model used on dataset 2. It consists +of a 512-neurons dense hidden layer and must be fed with a similarity to +function optimally. +4.2 +Clustering on the tweet vector +Once all the attributes related to popularity dynamics are gath- +ered, the tweet vector Π have the following form : +Π = (nmax +likes, slopemean, t10, t50, t90, n1 +boost, n2 +boost, n3 +boost, d1, d2, d3, d4) +where ni +boost is the raw increase of the ith boost and dj is the jth +component of the reduced L1 distance matrix for the current +tweet. +Now, the HDBSCAN algorithm is applied to the tweet +vector — the clustering distance being the euclidean dis- +tance between the components — in order to obtain better +results than before (section 3.3). The clustering parameters +min_cluster_size and min_sample are set to the same val- +ues as previously (i.e. 2) to allow comparison between the two +methods. + +3 000 +512 +reconstructed +L1 +10000 +Latent +Input +Intermediate +Intermediate +Output +Space +Extracted11 +Figure 18. Encoding and identification of the boosts (λ = 0, 6). + +12 +Tweets’ popularity dynamics +4.3 +Results +Table 2. Summary of the tweet vector clustering results +Dataset +1 +2 +Both +Number of tweets +2785 +3001 +5786 +Number of clusters +341 +329 +690 +Noise rate +43,7% +41,0% +42,6% +Average cluster size +8 tweets +9 tweets +8 tweets +Standard +deviation +of the clusters size +65 tweets +68 tweets +93 tweets +The 12 first clusters for both dataset 1 and 2 are displayed in +Appendix 3. +Clearly, the use of the tweet vector hasn’t improved the +clustering significantly. Worse, it has probably even degraded +it. Indeed, the number of clusters has not reduced — the value +exposed here is lower but it must be kept in mind that the iter- +ative clustering has not been applied — and their composition +is sometimes surprising (see the 7th cluster). +Nevertheless, the boost detection tool remains helpful to +characterize the different evolutions. +5. +OVERALL ANSWERS AND AVENUES FOR REFLEXION +In the end, this study have not allowed us to find a reasonable +number of easily interpretable patterns in tweets popularity +evolution. The naive approach, which interpreted the dynam- +ics closeness as a point to point distance, enabled to identify +the main difficulties while providing an initial overview of +the problem. Thanks to this fast and simple construction, the +importance of accurately defining the elements manipu- +lated and properly quantifying the expected results has +been understood. The revision that followed tried to satisfy +those new requirements through the use of a tweet vector, whose +components are said to characterize the popularity dynamics. +Unfortunately, this sort of "feature extraction" didn’t lead to +encouraging results. +However, this strategy is far from being fully explored. +On the one hand, the effect of min_cluster_size and +min_samples has been underestimated. A solid method to +choose them unambiguously would be of great interest since +they exercise strong control over the amount of clusters. On +the other hand, the tweet vector features are likely to be in- +appropriate : a more elaborate work focused on the boosts or +similar objects may bring promising outcomes. +Whether or not a positive response can be provided to the +initial problem therefore depends essentially on the ability to +identify the right features on which the clustering algorithm +is applied. +References +[1] +Sahar ofcial website. 2022. 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URL: https: +//pytorch.org/docs/stable/generated/torch.nn.SmoothL1Loss. +html#torch.nn.SmoothL1Loss (visited on 08/28/2022). +Appendix 1. +Keywords and users associated to datasets +Dataset 1 : +Keywords +Hashtags +Users +Ukraine +#TopGun +@EmmanuelMacron +France +@elonmusk +Foot +@Thom_astro +Dataset 2 : +Hashtags +Users +#canicule +@fetemusique +#Ukraine +@NUPES_2022_ +#Sievierodonetsk +@top14rugby +#Luhansk +@KyivIndependent +#legislatives2022 +@ActuFoot_ +#Législatives +@AP +#F1 +@elonmusk +#CanGP +@AllanBARTE +#CanadaGP +@netflix +#FormulaOne +@NASA +#MontrealGP +#Taiwan +#FathersDay +#FeteDeLaMusique +#NUPES +#LREM +#Climate +#ClimateEmergency +#ClimateChange +#HumanRights +#RefugeesDay +#Glastonbury2022 +#glastonburyfestival +#Bitcoin +#EndGunViolence +#KevinSpacey +#COVID19 +Appendix 2. +Clustering results with the naive approach +Appendix 3. +Tweet vector clustering results + +14 +Tweets’ popularity dynamics +Figure 20. The first 18 clusters (sorted by nmax +likes) obtained from dataset 1 : dmoy +L1 represents the average L1 distance within each cluster, n_boostsmean stands for +the average amount of boosts (cf section 4.1.2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared. The red line +is the average of all the gray curves (which are the actual time series). + +15 +Figure 21. The first 18 clusters (sorted by nmax +likes) obtained from dataset 2 : dmoy +L1 represents the average L1 distance within each cluster, n_boostsmean stands for +the average amount of boosts (cf section 4.1.2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared. The red line +is the average of all the gray curves (which are the actual time series). + +16 +Tweets’ popularity dynamics +Figure 22. The first 18 clusters (sorted by nmax +likes) obtained from both dataset 1 and 2 : dmoy +L1 represents the average L1 distance within each cluster, n_boostsmean +stands for the average amount of boosts (cf section 4.1.2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared. +The red line is the average of all the gray curves (which are the actual time series). + +17 +Figure 23. The first 12 clusters (sorted by nmax +likes) obtained from the tweet vectors of both dataset 1 and 2. + diff --git a/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/load_file.txt b/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e70dbc9275759a6e33ac23d24aae7fac96d60f6 --- /dev/null +++ b/S9AyT4oBgHgl3EQf8Ppy/content/tmp_files/load_file.txt @@ -0,0 +1,734 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf,len=733 +page_content='(2022), 1–17 R&D INTERNSHIP REPORT Tweets’ popularity dynamics Ferdinand Willemin Sahar [1], France Email: ferdinand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='willemin@ecl20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='ec-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 28th August 2022 Abstract This article charts the work of a 4 month project aimed at automatically identifying patterns of tweets’ popularity evolution using Machine Learning and Deep Learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To apprehend both the data and the extent of the problem, a straightforward clustering algorithm based on a point to point distance is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Then, in an attempt to refine the algorithm, various analyses especially using feature extraction techniques are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Although the algorithm eventually fails to automate such a task, this exercise raises a complex but necessary issue touching on the impact of virality on social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Keywords: IA, TIME-SERIES, TWITTER, VIRALITY, CLUSTERING, HDBSCAN, BIG DATA, TV DENOISING 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 Context and stakes Sahar is a private company specialized in collecting, processing and visualizing massive open-access data available in the web, including social networks data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The firm provides a data anal- ysis tool tailored to very different types of clients, requiring it to be both exhaustive and flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This work aims to comprehend popularity mechanisms within social networks in an attempt to help improve some of Sahar’s tool features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Nowadays, Twitter is the ultimate medium for informa- tion broadcasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It encompasses more than 200 millions usersa around the world and is likely to serve both individuals as well as institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Moreover, its recent effects over politics and the economy have placed it at the core of global attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This is why we picked it to study popularity mechanisms on social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 Twitter Twitter [2] is a social network, and more specifically a micro- blogging service, where users can post and interact with messages known as tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' A user can interact with a tweet in three different ways : he can comment it, retweet it or like it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To comment a tweet is to publicly answer a tweet, with a message that will appear in the tweet’s comment section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To retweet means to display a tweet on one’s own profile, in order to share it with one’s followers (people that are aware of your activity on the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To like is to show one’s consensus and/or support to a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Tweets are limited to 280 char- acters which makes them as much easily broadcastable as highly ephemeral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Tweets often come with hashtags : key- words preceded by the typographic sign # added with the aim ahttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='blogdumoderateur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='com/chiffres-twitter/ of specifying themes the tweet resonates to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Screen shot of a tweet and its interaction buttons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [3] The manner in which Tweets spread across the network — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' their way to be shared or seen by users — is atypical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' That is why a study dedicated to them is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Definition of the problem The problem can be formulated as follows : Can a reasonable number of interpretable patterns showing the dynamics of tweets’ popoularity through time be identified ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In order to provide a sufficient answer to the above prob- lematic, certain subjective terms must be defined and precised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The exploratory (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' being solely results-driven) nature of this work has opened to door to the following precisions: the pop- ularity is arbitrarily designated by the number of likes, a rea- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='00853v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='LG] 2 Jan 2023 CliffSchecter @cliffschecter As a reminder we have zero proofZEROgun laws work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content="Youknow,if you don't include Japan,U." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Belgium,Canada,Iceland,Romania,Norway,Austria Argentina,Netherlands,SouthKorea,Italy,Greece, Chile,France,Spain,Sweden,Singapore,Portugal, Israel,Czechia,Denmark, 1:38AM·May25,2022+TwitterforiPhone 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='8KRetweets 1,184QuoteTweets 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4K Likes Don 7 C [Like Tweet your reply Reply2 Tweets’ popularity dynamics sonable number is likely to be below 20 and interpretable patterns must be distinct enough so that we can name and describe them unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Note : In this work, we treat popularity and virality to be the strictly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4 State of the art Research on this topic emanates from the rise of social net- works which is a relatively recent phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The year 2008 can be considered the representative year of this ten- dency, as it constitutes Facebook’s passing over 100 million users [4], making it the first social network at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Twitter reached this count in 2011 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first noticeable study dealing with popularity dynam- ics in user-generated content [6] is applied to the video host YouTube, with the aim of finding locally relevant content dif- fering from the well-known most popular content that only considers mass appeal and an instantaneous vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The analysis leads to the identification of three categories : the junk, the quality and the viral video dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Junk videos undergo a burst of popularity which drops quickly afterwards because they do not spread through the social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Quality videos meet a very sudden peak in popularity, certainly caused by an exogenous effect (such as being featured on the first page of YouTube), and a subsequently passive decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Viral videos face a slowly increase up to a peak followed by a slow de- crease which reflects a word-of-mouth process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Concurrently, most of the videos do not experience any peak in popularity, embodying a fourth category : the silent videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' YouTube is actually not a social network but some parallels can be drawn between its features and Twitter’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' For instance, the social network displays, alike Youtube, the trendiest topics on its first page which can trigger exogenous effects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, apart from the difference of contents’ nature be- tween the two websites, the article [6] is based on applying a mathematical model (the self-excited Hawkes conditional Poisson process) to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The assumptions made to use it are not detailed enough and may not fit with our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As a result, the origins of the four categories are not fully defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Another interesting work [7] implements a clustering al- gorithm to gather hashtags’ popularity dynamics with similar shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The popularity is measured by the number of appear- ances of a hashtag over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In summary, it uses a K-Means algorithm provided by an "improved" euclidean distance that ignores both the overall hashtags’ appearances and the temporal gap between the popularity major peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Although the idea of using a clustering algorithm may seem appropriate — since such algorithms are used to classify objects according to specified features — some of the choices made by the author hide some conjectures that are arguable in the context of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' First, given two tweets and the history of their number of likes, does having the same shape but at different scales with a time lag can be deemed to be similar dynamics ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 2 illustrates this issue by showing two evolutions whose shapes could be considered as similar as they both include successively a quick growth, an effective stage, a slower but longer rise and eventually a cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, their maximum number of likes (nmax likes) vary with a factor of almost 30 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The processes that generated them do not engage the same range thus their dynamics are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The use of a K-Means algorithm raises other questions, especially around the choice of the K hyperparameter which sets the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Moreover, the preprocessing adopted —implying to manipulate the 1000 most frequently mentioned hashtags — does not guarantee to lead to a large enough dataset where all patterns possible are represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Two tweet’s history of likes designated by their Twitter id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [8] also makes use of a clustering strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' They build a two-dimensional feature space and a correlation- based similarity metric combined with the affinity propagation algorithm [9] to extract the main evolution patterns from their datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Their approach is worth exploring but its complexity — especially the simultaneous analysis of two distant measures carrying information of different nature — makes it a method for further investigations that should be implemented at a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Unfortunately, time did not allow our study to experi- ment with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' A different way of seeing things, presented by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Ouyang [10], represents the evolution of videos’ popularity from the chinese service provider Youku with a succession of two states : 1 if the video is experiencing a burst of popularity and 0 otherwise, the popularity corresponding to the number of views in a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' A burst stands for a striking increase in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Quantitatively, it is represented by an exceed- ing of a certain threshold by the derivative, whose value is actually established at three times the average derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The idea of extracting key information from curves to discrim- inate them is encouraging but requires additional work to be 3 carried out in order to gain in rigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Among others, this representation may only embrace sketchy tendencies and by doing so neglect some nuances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' For instance, integrating the size or duration of the different states could allow us to distinguish subgroups between them or might reveal some typical behaviors related to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' More specifically, the threshold’s value must come with a reliable justification that exhibit a true social phenomenon and the burst’s mathematical definition itself may need to be re- viewed to improve its robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' All in all, many lines of attack have been exposed over the previous years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' They should now be used to meticulously define a tailor-made strategy which will attempt to answer the initial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='5 Organization of the article Once the data presented (2), we will implement a HDB- SCAN clustering algorithm [11] specifically designed for our study (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Then, in an attempt to produce customizable and better results, another version of the HDBSCAN algo- rithm will be applied on a transformed dataset (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' By doing so, an original method created to reduce and store information of the popularity’s history into what is called a tweet vector will be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Ultimately, an overall assessment of the tech- niques developed followed by a list of promising avenues for further exploration of the topic will be proposed (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' PRESENTATION OF THE DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 The data fetching system Part of Sahar’s solution is an intelligent web scrapingb feature of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It notably enables users to analyze all messages containing one or several keywords — commonly a hashtag — and all the ones published by one or several given users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Thus, collecting an important chunk of messages around a particular subject is made extremely efficient and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' There are two main advantages to using a topic-oriented scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On the one hand, tracking all tweets posted in a given period of time would be too costly both in time and energy : around 500 millions tweets are produced each day [12] and assuming that their average lifetimec is a few days, each dataset would hold billions of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On the other hand, it is conceivable that evolutionary patterns will vary depend- ing on the topics addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Hence, Sahar’s tool’s filtering capability prevent us from mixing the topics — and therefore the patterns — too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Furthermore, let us recall that the project’s conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1) demand the virality algorithm to be compatible with the other features of Sahar’s tool, as they are to work in synergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' bAct of collecting publicly available data from the web, often on a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' cDuration starting when the message is published and ending when it no longer generates interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' From all the above, it only seems natural that we use the com- pany’s own scrapping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To ensure a significant diversity regarding the amplitude of the data monitored, the keywords or accounts followed were selected for their capacity to generate many likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' With- out this precision, we would automatically collect all tweets, especially noisy ones that do not generate any interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In effect, these inert tweets constitute the large majority of the publications [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Besides, by removing those inert tweets not only do we save time, but we also prevent our datasets from being excessively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 The resulting datasets Two datasets are used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first one, named dataset 1, is composed of 2785 tweets observed from 23/05/2022 to 27/05/2022 included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The points of each time series are recorded every 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The second one, dataset 2, is composed of 3000 tweets monitored from 17/06/2022 to 26/06/2022 included with a record every 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The keywords and users associated to each one of them are detailed in Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='B: It must be kept in mind that the frequency of data ac- quisition is not always constant in practice, due to the web scrapping feature performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Rudimentary preprocessing Figure 3 shows the evolution of popularity right after it had been collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The exact time is not specified to not overload the plot although it has a 1-second accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Typical raw data from the 1st dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This raw data format must undergo a first and obvious pre- processing (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To enable comparison between the different evolutions, the x axis is expressed in terms of duration in SI units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In the meantime, a large part of the asymptotic behavior is removed since it doesn’t carry any relevant infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To do so, we store the values t10 and t95 (respectively the times at which the curve reach 10% and 95% of its maxi- mum value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This draws an interval ∆t = t95 – t10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Then, we "extend" the interval by 20% by defining tmax = t10 + 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Finally, the data from 0 to tmax provides a window focused on the evolution dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4 Tweets’ popularity dynamics Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Example of the raw data and its elementary values (on top) trans- formed into a more suitable format (on the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' CLUSTERING Clustering belongs to the unsupervised class of machine learn- ing algorithm because it doesn’t involve two steps including a training stage where inputs are entered simultaneously as their corresponding outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On the contrary, it engages a single step where inputs are agglomerated to form clusters according to their distance to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Therefore, building a metric which represents how close the popularity evo- lutions are is fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Because it truely influences the quality of the results, it is the touchy phase of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To design such a model, it is essential to answer the follow- ing questions beforehand : what does having similar popularity dynamics means ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' and what is a good cluster in our case ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Through a naive approach, we answer the first problem by looking for a point to point proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The second issue is more tricky since it implies subjective criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' For instance, one may prefer to get homogeneous clusters regarding the number of tweets it contains without being demanding on the groups’ inner proximity, while another may want to obtain very accurate clusters, whatever the quantity of curves they hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To get around such complications, quality is arbitrarily favored, taking into account that our notion of proximity is by nature very demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In addition, the algorithm chosen in this study comes with a useful feature : the existence of a noise cluster, where all the unique dynamics — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' those that are "far" to every other — are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The size of this extra lot is also a criteria that can lead to new compromises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 Choice of the metric A prior search of the existing work conducted to the testing of two different distances : the Dynamic Time Warping distance and what we call the L1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first one naturally allows to avoid the main problem caused by the inaccurate recording of the data (cf 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2) : the disparity of the values along the time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The L1 distance doesn’t possess the same ability so a more elaborated preprocessing is needed to tackle this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, the latter is faster to compute and more respectful of the temporal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 Dynamic Time Warping distance The Dynamic Time Warping distance (or DTW distance) between two time series A and B is introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It is determined as follow : Let {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , ap} and {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , bq} respectively be the values of A and B along the Y axis (note that p and q don’t have to be equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Then, let W designate the set of all the paths between A and B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the selections of (i, j) where every points from A and B are involved at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Let also consider that a simple distance δ, typically the euclidean distance, has been settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' For all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , p} and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , q}, δ(ai, bj) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The results can be visualized in a distance matrix D = � di,j � 1≤i≤p 1≤j≤q where di,j = δ(ai, bj) (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The DTW distance is then defined as DTW(A, B) = min w∈W � � � (i,j)∈w δ(ai, bj) � � (1) The path w that lead to the DTW distance is called the warping path (see figures 5 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This is equivalent to applying a non-linear temporal distortion to the data which aligns on the time axis points that are the closest towards the Y axis and then return the sum of their distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Visualization of a fictional distance matrix and its warping path (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Many aspects of this definition are physically incoherent : for a given i, calculating δ(ai, bj) ∀j means that the tempo- ral dimension is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To address these obstacles, some correcting constraints are added : the boundary constraint stipulates that a path must include (1, 1) and (p, q) which guarantees to consider a beginning and end notion ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the monotonicity constraint states that ik ≤ ik+1 ∀k and jk ≤ jk+1 ∀k which compels the points inspection to al- ways advance in time ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' ~ 100% 95% 10%D = di, S(a1, b1 s(ai, b1) a15 the continuity constraint imposes that |ik+1 – ik| ≤ 1 ∀k and |jk+1–jk| ≤ 1∀k which forces comparing distances between neighbor points, and therefore to progress in both time series "at the same speed" ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the warping window specifies a maximum range (R) of points to visit to prevent the points scan to be "stuck" in one time series while it keep going in the other : it can be expressed as ∀k, |ik – jk| ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Illustration of the DTW correcting constraints and their efect on the algorithm : the monotonicity constraint in red, the continuity constraint ingreenandthewarpingwindowinpurple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Here,thepathcanonlycontinue to the green locations that are not in hatched zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Once these restrictions are employed, a recursive formula can be revealed to find a suitable warping path : γ(i, j) = δ(i, j) + min[γ(i – 1, j), γ(i – 1, j – 1), γ(i, j – 1)] (2) where γ(i, j) stands for the cumulative distance until point (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [14] also proposes an algorithm to determine this distance — the DTW algorithm — which runs in quadratic time (O(pq)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, an optimization of the correcting constraints com- bined with an appropriate reduction of the data (called Ab- straction approach) can result in an upgrade that works in linear time : the FastDTW [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The latter is used for our calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 "L1" Distance A more straightforward method consists in computing a point to point distance between the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, comparing them at similar instants requires to have their points aligned along the time axis and to carry the same amount of points, otherwise the distance cannot be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As mentioned before (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1), our data doesn’t satisfy these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To overpass this issue, the curves must be interpolated linearly with a fixed step [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To limit the loss of informa- tion caused by this process the step is set at 5min, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' twice as small as our recording frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To ensure that they Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Illustration of the warping path (black lines) between two tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' all possess the same number of points, they are extended by considering they have reached their asymptote : points equal to the last existing one are added until they get to the longest evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='d Conclusively, given n ∈ N∗ and two interpolated time series A and B with their respective points {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , an} and {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' , bn}, the distance is expressed by : dL1 = n � i=1 |ai – bi| (3) In its continuous form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' when step → 0), (3) is written dL1 = tmax � 0 |f – g|dx where f and g are the functions describing the two evolutions, which inspired the name L1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It coincides with the surface visible between the two curves (figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The lower this area is, the closer the dynamics are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Illustration of the L1 distance between two tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' dNote : With the benefit of hindsight, it would have been easier and more legitimate to modify the rudimentary preprocessing (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3) by truncating the asymptotes at the maximum tmax of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' R D= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='+ di+1,i+1 a1,6 di a1Msg n°1529195078606106625 Msg n°1529612363883810819Tweet n°1529612363883810819 Tweet n°15291950786061066256 Tweets’ popularity dynamics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Analysis of the distances Discriminating the metrics isn’t trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Indeed, their main goal is to bring the equivalent curves closer together and to move away from each other those that are different, so they must be judged on this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, they were built for this very purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The fact that they serve as both a test object and an evaluation instrument forces us to imagine alternative criteria of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Knowing that the human eyes are the best tools to execute this task for a low number of tweets, a qualitative survey is first conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' For each distance the ith closest pair of curves is examined and compared (i taking different values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Besides, some test samples are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Each one contains two pairs of tweets : one considered as close and the other as a distant pair according to human eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The goal is to see whether the pairs are labeled identically with our metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To make it happen, a random sub-sampling of the 1st dataset is implemented which selects 200 tweets out of 3278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This is necessary be- cause the DTW algorithm is quite time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As a reference, it takes about 5 seconds to calculate the pairwise L1 distance between the sub-sampled data (including the inter- polation step) while it takes about 30 minutes with the DTW distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Incidentally, this facet counts as a great prejudice against the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This inquiry reveals some weaknesses of DTW, as it some- times leads to doubtful results (figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On the contrary, the L1 distance does not experience such outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Even if a qualitative examination is not sufficient to con- clude, it exhibits another proof of DTW’s inferiority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As shown in figure 10, this distance is slightly influenced by the number of records present in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Despite some attempts to get rid of this drawback (mainly through a penal- ization strategy), it remains disturbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Bypassing the problem with preprocessing would be absurd since it constitutes the main strength of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' All in all, due to its better robustness, simplicity and speed, the L1 distance is chosen as the reference distance for the rest of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, DTW is worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Its inherent upsides may be useful for other applications and can be greatly improved in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As for the compu- tational time, one must notice that the algorithms used may not be fully optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4 Adding a weighting Afterwards, it has been assumed that the beginning of the popularity evolution was carrying more information about the underlying social dynamics than the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To introduce this feature into our metric, a weighting is added, favoring the first instants and handicapping the last ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Concretely, it implies multiplying the L1 distance with a penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Several functions were studied to eventu- ally select the one that offered the most control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It is defined as : f (t) = (1 – ϵ) ∗ �th(β – α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='t) + 1 2 � + ϵ (4) where : β is such that f (0) = 0, 99 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' α is such as f (tmax) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='7 with tmax being the median of all the tmax of the dataset ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='05 stipulating that the minimum weight is 5% The weighting’s influence can be seized in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Graph showing the impact of the weighting on the L1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Note : the amplitude of the weighting has been increased for the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 The clustering algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 HDBSCAN HDBSCAN [17] is originally a DBSCAN [18] upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Both are density-based algorithm, which means they have been conceived to identify dense areas within groups of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As mentioned before, one of their considerable advantages is their capacity to generate a noise cluster where all the unique dynam- ics are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Actually, all time series that are not in a dense zone are considered as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To execute such a task, it needs to be provided with two main parameters : min_samples and min_cluster_size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' min_samples is used to define a disk surrounding each data point whose ray corresponds to the distance between a point and its min_samples’s closest neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Two points whose L2 distance is inferior to both point’s rays will be con- sidered to be in the same dense area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' If only one of the two belongs to the disk of the other, their distance will be increased to match with the bigger ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Therefore, min_samples is used to redefine the space topology by accentuating den- sity phenomena: the points which are not in dense areas are even more isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The distance created by this process is called the mutual reachability distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' min_cluster_size corresponds to the minimum size for a dense area to be considered as a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Bellow this value the time series inside the zone are viewed as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To Tweetn°1529595180587929600 Tweetn°1529969772011610144 Weightingfunction7 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Two pairs of tweets extracted and their "proximity rank".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' According to the DTW distance, the lef pair contains closer tweets than in the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Qualitatively, one can judge the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Variation of the average DTW distance according to the number of points (raw plot on the lef, without extreme points on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It seems that time series carrying a lot of information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' many records) have a tendency to move away from the others relatively to the DTW distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' better seize its impact, a deeper comprehension of the algo- rithm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Let’s consider the minimum spanning tree of the dataset [19] built from the mutual reachability distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' By defini- tion, this tree links all the points together each time with a single bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Let’s consider that when a point is linked with another, they both form a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Let’s delete the bounds one by one from the biggest to the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Removing a bound is equivalent to divide a parent cluster into two children clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Therefore at the end, all the points are isolated from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' All the divisions triggered during the process can be visualized in a dendogram as in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Since this method leads to an enormous amount of clus- ters, a refinement is added : rather than seeing each split as a parent cluster giving birth to two children clusters, it could be interpreted as a "cluster erosion" : if a children is too small to be a cluster itself, it means that the parent cluster didn’t actually split, but instead lost some points that became noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Hence, min_cluster_size is there to define what too small means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 13 shows the kind of change provoked on the dendogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Dendogramdepictingclusters’splitsasϵ(herebeingnormalized and called distance) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [11] Correlationbetweennumberofpointsand DTWdistance 1e6 6 / distance to the othercurves 5 3 0 0 500 1000 1500 2000 2500 Amount of records withinthe time seriesCorrelationbetweennumberofpointsand DTWdistance 32000 curves 31900 other 31800 to the 31700 /distance 31600 31500 DTW 31400 Average 31300 31200 0 500 1000 1500 2000 2500 Amountof recordswithinthetimeseries0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='6 log(Number of points) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='0 distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='08 Tweets’ popularity dynamics Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The same dendogram afer applying min_cluster_size � λ = 1 ϵ � [11] Eventually, a flat clustering is extracted by keeping the most stable groups that don’t overlap with each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Crudely, they relate to the longest ones in the dendogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This stands for the basic knowledge necessary to under- stand our way of using HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' A more exhaustive and detailed explanation can be found in the library documentation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 Parameters selection Building a score to choose the parameters in a way that guar- antees both a reasonable number of clusters and a sufficient proximity within them was first intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Unfortunately, it didn’t succeed mainly because it was too sensitive : the score obtained after an important variation of the parameters — around 10 units — was so close that the "optimal" parameters could dramatically change between the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' These param- eters are so affecting that such a behavior cannot be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As a result, another indicator is adopted : the size of the noise cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Usually, this quantity is relatively large — from 40 to 50% of the dataset — which is quite annoy- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Hence, it is desirable to reduce it as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The best lowering is achieved with the lowest values of both min_cluster_size and min_samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It is not sur- prising at all knowing their influence on the clustering (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Considering that the accuracy of the clusters is fa- vored over a reasonable amount of groups (cf 3), setting both min_samples=2 and min_cluster_size=2 is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Ap- plying these values on the merged datasets (5786 tweets) triggers 45,37% noise (2625 tweets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Iterative clustering Even when the parameters are adjusted to minimize the noise rate, it remains really high (cf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Although knowing which time series have unique dynamics is important, such noise rate is excessive and may come from disproportionate proximity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To overcome that, an iterative clustering is pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Basically, each iteration consists in considering the noise cluster as a dataset itself on which the HDBSCAN algorithm is applied (figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' At each round the density notion is redefined since the number of inputs is lower, so new clusters emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As they are of poorer quality, the round in which they appeared is stored to be reminded when clusters are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Scheme representing the iterative clustering principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The iteration stops when the noise share is bellow 5% of the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Results In Appendix 2, the 18 first clusters obtained from each dataset are displayed (figure 20, 21 and 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Table 1 summarizes in- formation about the clusters obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Summary of the clustering results Dataset 1 2 Both Number of tweets 2785 3001 5786 Number of clusters 491 498 975 Noise rate 1,1% 1,0% 1,4% Average cluster size 6 tweets 6 tweets 6 tweets Standard deviation of the clusters size 9 tweets 14 tweets 17 tweets Highest cluster size 141 tweets 232 tweets 349 tweets Undoubtedly, these outcomes are not sufficient to answer our original problem (section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The reasonable number of clusters is far from being reached and some of them are too disparate to be clearly describable (see figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On top of that, clusters made of popular tweets — those with high values of nmax likes — are often too small to be considered as pattern repre- sentatives, although this is to be expected given our parameters selection (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Several causes could explain such poor results : 0 100 90 1 80 2 70 3 Number of points 60 lue va 4 50 40 5 30 9 20 7 10 8 0Noise cluster Dataset Clusters Clustering Noise cluster Dataset Clusters Clustering9 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Biggest cluster from the 1st dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the study might suffer from a lack of data : datasets might be too little for each pattern to be represented by an ap- propriate number of tweets ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' discovering a reasonable amount of interpretable patterns may not be possible because of the plurality of social dynamics ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' our definition of evolution dynamics itself (section 3) may be too extreme or at least not accurate : so far, two curves have the exact same dynamics if they are equal at any point (without considering the weighting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In the remainder of this article, a revision of this last point is proposed, as it seems to be the most obvious source of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' THE TWEET VECTOR Instead of measuring how close popularity evolutions are at any point, the study will now focus on the essential partic- ularities of what we call dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Then a clustering can be exclusively applied on those attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' As it corresponds to a relaxation of our previous distance measure, it may lead to a lower amount of clusters without compromising their quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Every element that can characterize a time series’ dynamics will be stored in a vector called the tweet vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 Components of the tweet vector The dynamics’ characteristics identified are : nmax likes — the maximum number of like — as it reflects the intensity of the dynamics ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' slopemean — the average slope of the curve — since it de- scribes how fast the dynamics are ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the key instants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the times at which the evolution reaches a given percentage of its maximum value, to take into account its temporal distribution ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the boosts’ raw increases (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2) which constitutes a striking distinction between evolutions ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' a compressed format of the L1 distance between the current curve and the others, which carries indications about its overall (and relative) shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Although nmax likes and slopemean are quite explicit values, the integration of the key instants, boosts and L1 distance com- ponents require some additional work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In doing so, we will attempt to obtain a tweet vector as small as possible, since the idea is to only carry necessary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1 The key instants It is assumed that extracting tp with p ∈ {10, 20, 30, 40, 50, 60, 70, 80, 90} (as manipulated is section 4) is more than enough to apprehend the temporal distribution of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The objective is to select the most relevant ones too reduce the tweet vector’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' To tackle this task, the well-known Prin- cipal Component Analysis (PCA) algorithm is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Details about its theoretical functioning can be found in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The PCA’s implementation of scikit-learn [21] allows to be aware the share of variance kept during the dimension reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It also offers the possibility to visualize a projection of the variance distribution of the different key instants over a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' After running it on the dataset, it turns out that retaining the 3 first components allows to keep 95% of the variance and that the less correlated triplet of key instants is (t10, t50 and t90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Although PCA’s components are not equals to the key instants themselves, we can suppose this particular triplet contains the information we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 The boosts A boost designates a sudden change of rhythm favoring an increase of popularitye (see figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It is a specific behavior of the virality evolutions which could become an efficient tool to describe a given pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Qualitative identification of boosts in a given popularity evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Instinctively, the mathematical definition must integrate a condition concerning the derivative of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' How- ever, data imperfections can make the task complicated (figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' eboosts are inspired by the bursts described in [10] Tweet n°1529236803160461312 6000 5000 4000 boosts 3000 slowdowns 2000 1000 end of evolution 0 0 20000 40000 60000 80000 100000 120000 140000 Time (s)10 Tweets’ popularity dynamics Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Raw derivative of the time series window : the second boost does not stand out enough to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' A consistent way to highlight boosts while smoothing the derivative is to employ Total Variation Denoising (TV De- noising).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This technique consists in approaching the deriva- tive with a piecewise constant curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' This is equivalent to solving the following problem : find Rmin = argmin (f (O, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=')) where f (O, R) = ∥O – R∥2 + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' ∥∇R∥1 Here, O is the original derivative values, R is the vector repre- senting the reconstructed derivative and λ is a scalar affecting the "penalty" applied to the variation ∇R : the higher λ is, the less variations the reconstructed curve contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In prac- tice, this parameter is chosen so that the maximum number of boosts in the dataset doesn’t exceed 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Beyond 6, some boosts are likely to be mistakes caused by an excessive sensitiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Thanks to the cvxpy python library [22], this method is easily implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Boosts correspond to the denoised derivative’s bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' In order to detect them properly, an encoding program is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' First, the variations of the denoised derivative are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' After removing the numerical noise, the signs of it are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Hence, each tweet is related to a vector made of 1, -1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Finally, the boosts are enumerated and identified knowing that they start with a 1 and end as soon as a -1 is en- countered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Moreover, if the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' the last) sign is -1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 1), it means the curve begins (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' ends) with a boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 18 illustrates the different steps of the process and spotlights the resulting boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The raw increase of boosts is added to the tweet vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Hence, 6 components should be dedicated to it with the ith component representing the ith boost of the evolution (or equal to 0 if there is no ith boost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, the 3 first boosts seem to be the most determining : a tweet having 4 boosts is often almost inert, so that a single like is considered as a great increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' That is why only three components are eventually allocated to this characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 The L1 distance integration To be integrated to the tweet vector, the L1 distance matrix — containing all the pairwise L1 distances of the dataset — is reduced with an auto-encoder neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' With- out detailing the fundamentals of deep learning, this model is built with two symmetrical networks : the encoder and the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first one is trained to encrypt the data in a lower dimensional space while the second one is trained to recon- struct the initial data based on the encryption (also called latent space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' If the decoder succeeds in retrieving the original data, it means the latent space carries the essential information of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The architecture of our model for the 2nd dataset is shown in figure 19 : for each tweet, 4 components are enough to describe its distance to all the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The ELU activation function [23] and smooth L1 loss function [24] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Scheme representing the model used on dataset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' It consists of a 512-neurons dense hidden layer and must be fed with a similarity to function optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 Clustering on the tweet vector Once all the attributes related to popularity dynamics are gath- ered, the tweet vector Π have the following form : Π = (nmax likes, slopemean, t10, t50, t90, n1 boost, n2 boost, n3 boost, d1, d2, d3, d4) where ni boost is the raw increase of the ith boost and dj is the jth component of the reduced L1 distance matrix for the current tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Now, the HDBSCAN algorithm is applied to the tweet vector — the clustering distance being the euclidean dis- tance between the components — in order to obtain better results than before (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The clustering parameters min_cluster_size and min_sample are set to the same val- ues as previously (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 2) to allow comparison between the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 3 000 512 reconstructed L1 10000 Latent Input Intermediate Intermediate Output Space Extracted11 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Encoding and identification of the boosts (λ = 0, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 12 Tweets’ popularity dynamics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='3 Results Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Summary of the tweet vector clustering results Dataset 1 2 Both Number of tweets 2785 3001 5786 Number of clusters 341 329 690 Noise rate 43,7% 41,0% 42,6% Average cluster size 8 tweets 9 tweets 8 tweets Standard deviation of the clusters size 65 tweets 68 tweets 93 tweets The 12 first clusters for both dataset 1 and 2 are displayed in Appendix 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Clearly, the use of the tweet vector hasn’t improved the clustering significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Worse, it has probably even degraded it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Indeed, the number of clusters has not reduced — the value exposed here is lower but it must be kept in mind that the iter- ative clustering has not been applied — and their composition is sometimes surprising (see the 7th cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Nevertheless, the boost detection tool remains helpful to characterize the different evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' OVERALL ANSWERS AND AVENUES FOR REFLEXION In the end, this study have not allowed us to find a reasonable number of easily interpretable patterns in tweets popularity evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The naive approach, which interpreted the dynam- ics closeness as a point to point distance, enabled to identify the main difficulties while providing an initial overview of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Thanks to this fast and simple construction, the importance of accurately defining the elements manipu- lated and properly quantifying the expected results has been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The revision that followed tried to satisfy those new requirements through the use of a tweet vector, whose components are said to characterize the popularity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Unfortunately, this sort of "feature extraction" didn’t lead to encouraging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' However, this strategy is far from being fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' On the one hand, the effect of min_cluster_size and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' HuffPost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Section: Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 12, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='huffpost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [22] CVXPY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2 documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='cvxpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' org/ (visited on 08/28/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [23] ELU — PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='12 documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' URL: https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='ELU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='html (visited on 08/28/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' [24] Smooth L1 Loss — PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='12 documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' URL: https: //pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='SmoothL1Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' html#torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='SmoothL1Loss (visited on 08/28/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Keywords and users associated to datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Dataset 1 : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Keywords ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Hashtags ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Users ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Ukraine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='#TopGun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='@EmmanuelMacron ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='#Bitcoin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='#EndGunViolence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='#KevinSpacey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='#COVID19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Clustering results with the naive approach Appendix 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' Tweet vector clustering results 14 Tweets’ popularity dynamics Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first 18 clusters (sorted by nmax likes) obtained from dataset 1 : dmoy L1 represents the average L1 distance within each cluster, n_boostsmean stands for the average amount of boosts (cf section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The red line is the average of all the gray curves (which are the actual time series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 15 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first 18 clusters (sorted by nmax likes) obtained from dataset 2 : dmoy L1 represents the average L1 distance within each cluster, n_boostsmean stands for the average amount of boosts (cf section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The red line is the average of all the gray curves (which are the actual time series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 16 Tweets’ popularity dynamics Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first 18 clusters (sorted by nmax likes) obtained from both dataset 1 and 2 : dmoy L1 represents the average L1 distance within each cluster, n_boostsmean stands for the average amount of boosts (cf section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content='2) in each cluster and ROUND symbolizes the iteration number during which the cluster appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The red line is the average of all the gray curves (which are the actual time series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' 17 Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} +page_content=' The first 12 clusters (sorted by nmax likes) obtained from the tweet vectors of both dataset 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQf8Ppy/content/2301.00853v1.pdf'} diff --git a/S9E1T4oBgHgl3EQfIANs/content/tmp_files/2301.02933v1.pdf.txt b/S9E1T4oBgHgl3EQfIANs/content/tmp_files/2301.02933v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa83eeae236421d819d4a04287ef564207804f41 --- /dev/null +++ b/S9E1T4oBgHgl3EQfIANs/content/tmp_files/2301.02933v1.pdf.txt @@ -0,0 +1,2335 @@ +Weakly Supervised Joint Whole-Slide Segmentation and Classification +in Prostate Cancer +Pushpak Pati†1∗, Guillaume Jaume†2,3,4,5, Zeineb Ayadi8, Kevin Thandiackal1,6, +Behzad Bozorgtabar7, Maria Gabrani1, Orcun Goksel6,9 +1IBM Research Europe, Switzerland +2Department of Pathology, Brigham and Women’s Hospital, Harvard Medical School, USA +3Department of Pathology, Massachusetts General Hospital, Harvard Medical School, USA +4Cancer Program, Broad Institute of Harvard and MIT, USA +5Data Science Program, Dana-Farber/Harvard Cancer Center, USA +6Computer-Assisted Applications in Medicine, ETH Zurich, Switzerland +7Signal Processing Laboratory 5, EPFL, Switzerland +8EPFL, Switzerland +9Department of Information Technology, Uppsala University, Sweden +Abstract—The segmentation and automatic identification of +histological regions of diagnostic interest offer a valuable aid +to pathologists. However, segmentation methods are hampered +by the difficulty of obtaining pixel-level annotations, which +are tedious and expensive to obtain for Whole-Slide images +(WSI). To remedy this, weakly supervised methods have been +developed to exploit the annotations directly available at +the image level. However, to our knowledge, none of these +techniques is adapted to deal with WSIs. In this paper, +we propose WHOLESIGHT, a weakly-supervised method, +to simultaneously segment and classify WSIs of arbitrary +shapes and sizes. Formally, WHOLESIGHT first constructs +a tissue-graph representation of the WSI, where the nodes and +edges depict tissue regions and their interactions, respectively. +During training, a graph classification head classifies the WSI +and produces node-level pseudo labels via post-hoc feature +attribution. These pseudo labels are then used to train a node +classification head for WSI segmentation. During testing, both +heads simultaneously render class prediction and segmentation +for an input WSI. We evaluated WHOLESIGHT on three +public prostate cancer WSI datasets. Our method achieved +state-of-the-art weakly-supervised segmentation performance +on all datasets while resulting in better or comparable classifi- +cation with respect to state-of-the-art weakly-supervised WSI +classification methods. Additionally, we quantify the general- +ization capability of our method in terms of segmentation and +classification performance, uncertainty estimation, and model +calibration. +Keywords-Computational Pathology, Whole-Slide image seg- +mentation, Weakly supervised learning, Weakly supervised +classification, Weakly supervised segmentation +I. INTRODUCTION +Prostate cancer is the second most frequently diagnosed +cancer in men in the United States, with 250,000 new reg- +istered cases resulting in 35,000 deaths in 2022 [1]. Yet the +*Corresponding author: Pushpak Pati. Email: pus@zurich.ibm.com +number of pathologists, whose role is critical in the diagnosis +and management of cancer patients, is gradually declining. +In the United States, an 18% decrease was recorded between +2007 and 2017, resulting in a 42% increase in the average +workload [2]. In addition, the practice of uro-pathology also +has its share of challenges [3]. Although the diagnostic +criteria for grading prostate cancer are established [4], the +continuum of phenotypic features across the diagnostic spec- +trum leaves room for disparities, with significant intra- and +interobserver variability [5], [6]. The manual inspection of +slides is also tedious and time-consuming, and would benefit +from automation and standardization. These elements justify +the development of Computer-Aided Diagnosis (CAD) tools +to automate the diagnostic workflow. +To this end, several Artificial Intelligence (AI) based CAD +tools are proposed, including nucleus segmentation and clas- +sification [7]–[9], gland segmentation [9]–[11], and tumor +detection [12], [13]. Albeit the remarkable performance, +these tools often demand task- and tissue-specific annota- +tions on large datasets, which are tedious, time-consuming +and often infeasible to acquire. For reducing annotation +requirements, different approaches are proposed, in particu- +lar, weakly-supervised methods based on Multiple Instance +Learning (MIL) framework for the automatic classification +of Whole-Slide Images (WSIs) [14], [15]. +Although classification is useful, it remains limited in +its role of supporting the pathologist’s attention during +diagnosis. In this context, semantic segmentation methods +are preferable as they enable the generation of pixel-level +delineation of the tissue constituents that can highlight +diagnostically relevant regions. Such visualization allows +for strengthening trust between pathologists and CAD tools. +Additionally, the identified regions can be leveraged by a +classifier to improve patient diagnosis. However, semantic +1 +arXiv:2301.02933v1 [cs.CV] 7 Jan 2023 + +segmentation generally requires pixel-level labels, which +makes it more demanding in terms of annotations than +classification tasks. For this reason, the development of +weakly-supervised semantic segmentation (WSSS) methods +appears as the most adequate response. +While WSSS has been successful on natural images, it en- +counters various challenges when applied to histopathology +images [16], as they, (1) contain fine-grained objects with +large intra-class variations [17]; (2) often include ambiguous +boundaries among histology components [18]; (3) can be +several giga-pixels with arbitrary tissue sizes. Nevertheless, +some WSSS methods are proposed for various histology +tasks. The methods by [19]–[25] performing WSSS at patch- +level are limited by the need for patch-level annotations, +and inability to perform global contextualized WSI seg- +mentation. While [26], [27] scale to larger tiles, they pose +high computational complexity and memory requirements +for operating on WSIs. The methods by [25], [26] require +exact tile annotations for model training, i.e., a precise +denomination of each lesion type in a tile, which requires +pathologists to annotate images beyond standard clinical +needs. On a different note, recent WSI classification methods +use attention mechanisms or feature attributions to highlight +salient regions [14]. Though these regions are informative +for visual assessment, they are insufficient, incomplete, and +blurry for accurately delineating relevant regions. Addition- +ally, producing granular saliency requires densely overlap- +ping patch predictions, which is computationally expensive +while working with WSIs. +In +view +of +the +aforementioned +limitations +of +the +WSSS methods, we propose WHOLESIGHT, “Whole- +slide SegmentatIon using Graphs for HisTopathology”, +that can simultaneously segment and classify arbitrar- +ily large histopathology images by using WSI-level la- +bels, and without any task-specific assumptions or post- +processing. Formally, WHOLESIGHT transforms an im- +age into a superpixel-based tissue-graph (TG), and con- +siders the segmentation problem as a node-classification +task. WHOLESIGHT incorporates both local and global +tissue microenvironment to perform contextualized segmen- +tation, principally in agreement with inter-pixel relation- +based WSSS [28]. To summarize, our contributions are: +• WHOLESIGHT, +a +novel +graph-based +weakly- +supervised method to jointly segment and classify +WSIs using readily available WSI-level annotations. +• A comprehensive evaluation of WHOLESIGHT on 3 +prostate cancer datasets for Gleason pattern segmen- +tation and Gleason grading, and benchmarked against +state-of-the-art WSI-level weakly-supervised methods. +• Thorough +generalizability +quantification +of +WHOLESIGHT on in- and out-of-domain cohorts in +terms of segmentation and classification performance, +uncertainty +estimation, +and +calibration +of +neural +network predictions. +A preliminary version of this work was presented as [29]. +Our substantial extensions herein include, (1) an improved +WHOLESIGHT method in terms of model architecture and +automatic synthesis of node labels, (2) extensive evaluations +on large cohorts of WSIs (approximately 100×), and (3) +generalization assessment. +II. RELATED WORK +A. Weakly-supervised histopathology image classification +Weakly-supervised classification of WSIs has been mostly +developed around MIL. In MIL, a WSI is first decomposed +into a “bag” of patches and are encoded by a neural encoder, +e.g., a Convolutional Neural Network (CNN). Then, an +aggregator pools the patch embeddings to produce a slide- +level representation for mapping to a class label via a neural +predictor. The aggregator can be based on an attention +mechanism weighing the importance of each patch, as in +[14], [30], or as recently proposed, it can take the form +of a transformer [31], [32] or a Graph Neural Network +(GNN) [33], enabling modeling inter-patch dependencies +and global context. Differently, context can be modeled +using multi-scale representations of WSIs, either via multi- +magnification patch embeddings [23], [34] or by learning to +automatically select important regions, as proposed in [35], +[36]. Despite the success of these approaches, they cannot +directly be extended for semantic segmentation. +B. Weakly-supervised histopathology image segmentation +WSSS approaches in histopathology can be categorized +by the type of supervision (or annotation), e.g., point an- +notations, scribbles, or image-level labels, and the scale of +operation, e.g., patches, tiles, Tissue-Micro Array (TMAs), +or WSIs. [37]–[39] utilized point annotations to segment +cells and nuclei in histology patches. [23], [40] used scribble +annotations to segment tissue and tumor regions, respec- +tively, at patch-level. Both the approaches used U-Net [41], +where [23] leveraged concentric patches across multiple +magnifications for including relevant context information, +and [40] modified the objective function to balance the +contribution of the annotated pixels. The majority of the +WSSS methods in histopathology utilized image-level su- +pervision and are limited to operate with patch annotations. +[19] proposed multiple clustered instance learning to process +sliding patches for simultaneous grading and segmentation +of colon TMAs. [21] trained a binary classifier for pixel- +level predictions and afterward computed an image-level +prediction from pixel labels via a softmax function. They +optimized image prediction, such that pixel predictions were +improved. [22] proposed CAMEL, a MIL-based label en- +richment method. It split an image into latticed instances, +generated instance labels, and assigned instance labels to +corresponding pixels to enable supervised segmentation. [26] +proposed HistoSegNet, which trained a CNN to predict +tissue types in a tile and used feature attribution to derive +2 + +pixel-level predictions. It also employed a series of dedicated +post-processing steps for prediction refinement. [24] used +foreground proportion as the weak labels and combined +a fully convolutional network and a graph convolutional +network for tissue segmentation. [25] proposed a feature +attribution-based model to generate pseudo labels, followed +by a multi-layer pseudo-supervision network for segmenting +tissue types. As a main limitation, these methods cannot +perform WSSS on WSIs using only WSI labels. To perform +WSSS beyond patch-level, [27] proposed WeGleNet, that +scales to TMAs. It included a segmentation- and a global- +aggregation layer to classify images during training, and up- +sampled pixel-level softmax activations during inference for +image segmentation. However, the method cannot precisely +delineate lesions and highlight multiple lesion occurrences. +It also requires processing densely overlapping patches for +fine segmentation, and cannot scale to WSIs. In contrast, our +WHOLESIGHT can perform WSSS by leveraging image- +level supervision, while efficiently scaling to WSIs of arbi- +trary dimensions. +C. Generalization quantification in histopathology +Generalizability of CAD tools in histopathology is af- +fected by domain-level biases, which are introduced due +to numerous reasons, such as different staining protocols, +manufacturing devices, materials, and scanning devices with +respective color response [42]. Though generalizable tools, +that are robust to domain shifts, are desired, it is challenging +to model and detect the domain shifts in Deep Learning +(DL). Nevertheless, several approaches have been proposed +to reduce such domain shifts via data- and model-level +adaptation. +Data-level adaptation can be achieved via stain normal- +ization [43]–[46], color augmentation [47], [48], or stain +invariant feature learning [49], [50]. Model-level adaptation +is typically done via domain adversarial training [51]–[54], +which leverages target domain unlabeled data along with +source domain data for modeling. +However, the aforementioned data- and model-level adap- +tation approaches do not exhaustively assess the gener- +alization ability of their trained DL models beyond task +performance. In this case, accurate uncertainty estimation +and model calibration are crucial to know when to trust the +model – a task known to be challenging for neural networks +that often provide over-confident predictions [55]–[57]. To +the best of our knowledge, computational pathology research +in these directions is scarce and remains unexplored. +III. METHODOLOGY +In this section, we present WHOLESIGHT for scalable +WSSS of histopathology images. First, we transform a WSI +into a TG representation, where nodes and edges of the +graph denote tissue regions and their interactions, respec- +tively (Section III-B). Next, a GNN contextualizes node +embeddings characterizing tissue regions (Section III-C), +which are then processed by a graph classification head +for Gleason grading (Section III-D). Finally, we generate +node-level pseudo labels using feature attribution and a node +selection strategy, which are used to train a node classifica- +tion head. The node-head outputs the segmentation mask +with pixel-level Gleason pattern assignment (Section III-E). +An overview of the method is presented in Figure 1. +A. Notation and preliminaries +We define a graph G ∈ G as (VG, EG, H), where VG and +EG denote the set of nodes and edges, respectively, H ∈ +R|V |×d denotes d-dimensional node features (or denoted at +node-level as Hv,. := h(v) ∈ Rd), and G is the set of graphs. +The neighborhood of a node v ∈ VG is denoted as N(v) := +{u ∈ VG | (v, u) ∈ EG ∨ (u, v) ∈ EG}. We represent the +cardinality of a set as |.|, e.g., |N(v)| indicates the number +of neighbors of v. +GNNs [58] are a class of neural networks that learn from +graph-structured data. Specifically, GNNs follow a two- +step procedure to contextualize node features by including +neighborhood node information. First, in an AGGREGATE +step, for each node v ∈ VG, the neighboring node features +N(v) are aggregated by a differentiable and permutation- +invariant function. Next, in an UPDATE step, the current +features of v and the aggregated features of N(v) are +processed by a differentiable operator to update the features +of v. This procedure is repeated T times, where T is the +number of GNN layers. +In this work, we use the Graph Isomorphism Network +(GIN) [59], where the AGGREGATE step is a mean-operator, +and the UPDATE step includes a multi-layer perceptron +(MLP). Formally, a GIN layer is given as, +h(t+1)(v) = MLP +� +h(t)(v) + +1 +|N(v)| +� +u∈N (v) +h(t)(u) +� +(1) +T GIN layers, denoted as Fθ, are stacked to acquire context +information up to T-hops for each v. For graph classi- +fication, a fix-sized graph-level embedding hG is derived +by pooling the node embeddings hT (v), ∀v ∈ VG by a +READOUT step, e.g., a mean-operation. Subsequently, hG +is mapped to target classes by a classifier network, Fφ. +Similarly, for node classification, hT (v), ∀v ∈ VG can be +classified by a classifier network Fψ. +Formally, classification aims to predict target label y ∈ K +for an input x ∈ X, where K and X denote the set +of classes and inputs, respectively. Given a set of sample +pairs {(xi, yi)}N +i=1, where N is the number of samples and +(xi, yi) ∼ p(x, y), the data likelihood can be expressed as +p(Y |X, θ, φ) = ΠN +i=1p(yi|xi, θ, φ). The optimal parameters +(ˆθ, ˆφ) are obtained by maximum likelihood estimation, or +equivalently by minimizing the Negative Log-Likelihood +(NLL) − �N +i=1 log p(yi|xi, θ, φ). For graph classification, +a sample pair is denoted as (yG, G), yG ∈ KG, G ∈ +3 + +Figure 1: Overview of the proposed WHOLESIGHT method. (a) In the preprocessing step, a TG is constructed to represent +a WSI, where the nodes and edges are defined by identifying superpixels and region adjacency connectivity, respectively. +(b) The graph classification head classifies the TG into primary and secondary Gleason patterns. Subsequently, a feature +attribution technique and a node selection strategy derive node-level pseudo-labels. (c) The node classification head learns +on the pseudo-labels to classify the nodes, thereby resulting in the WSI segmentation. +G. For node classification, a sample pair is denoted as +(yV , v), yV ∈ KV, v ∈ V. For the task in this paper, +the set of graph- and node-level classes are the same, i.e., +K := KG = KV. +We further introduce the notion of model calibration [60]. +Intuitively, the probability of outcomes, i.e., confidence +scores, of a calibrated model should match its performance. +For example, the samples predicted with an average confi- +dence of 60% by a model should have an average accuracy +of 60%. Formally, for a given network, f : X +→ K, +and p(X, Y ) a joint distribution over the data and the +labels, f(x) is said to be calibrated with respect to p if, +Ep[Y |f(X) = β] = β, ∀β ∈ [0, 1]. The calibration can +be visualized with a reliability diagram [61]. Namely, all +the samples in the dataset are assigned to bins according to +their predicted confidence scores. Then, the model accuracy +is computed for the samples in each bin. The network +performance is plotted against the binned confidence scores, +where deviations from the diagonal represent uncalibrated +bins. +B. Preprocessing and tissue-graph construction +First, we stain-normalize the input H&E stained images +using the method by [44]. It reduces appearance variability +across images caused during tissue preparation, i.e., different +specimen preparation techniques, staining protocols, fixation +characteristics, and imaging device characteristics [62], [63]. +Then, we transform the normalized images into TGs (Fig- +ure 1(a)), where the nodes and the edges of a TG denote +tissue regions and inter-tissue interactions, respectively. Mo- +tivated by [64], [65], we consider superpixels as the visual +primitives to encode tissue regions. Compared to rectangular +patches, superpixels are more flexible to accommodate ar- +bitrary shapes according to the local homogeneity of tissue. +The homogeneity constraint also restricts the superpixels to +span across multiple distinct structures and include different +morphological regions. +TG construction follows [66], where the prominent steps +are, (1) detection of superpixels to define nodes VG, (2) +characterization of superpixels to define node features H, +and (3) building graph topology to define edges EG. We +4 + +WSI +Superpixels +Tissue-graph +P+S: 3+4 +classification head +Primary +Attribution +Pseudo-labels +REA- +Benign +(b) Graph +G3 +G4 +GNN +MLP +G5 +D +Fe +Secondary +F +Benign +Benign +U +G3 +G3 +T +G4 +G4 +G5 +high +G5 +ow +Segmentation +classification head +(c) Node +GNN +MLP +Fe + Benign +G3 +G4 +G5 +Training +Frozen +Tissue-graph input +Graph supervision +Node supervisionadopt a two-step process to identify superpixels in a WSI. +First, we use Simple Linear Iterative Clustering (SLIC) [67] +to produce over-segmented superpixels. Over-segmentation +is conducted at a low magnification to capture homogeneous +regions while offering a good compromise between granu- +larity and smoothing-out noise. In the second step, the over- +segmented superpixels are hierarchically merged according +to their channel-wise color similarity at high magnification. +Color similarity is quantified in terms of channel-wise 8-bin +color histograms, mean, standard deviation, median, energy, +and skewness. The resulting merged tissue regions form the +nodes of the TG. The merged superpixels denote morpholog- +ically meaningful homogeneous regions. Additionally, merg- +ing reduces the node complexity of the TG, thus enables the +scaling of TG to a large WSI and contextualization to distant +tissue regions. +We characterize the TG nodes by morphology and spatial +features. Considering the potentially arbitrary dimension of +superpixels, we use a two-step process to derive morphology. +First, we extract patches of size 144×144 pixels from a su- +perpixel, resize them to 224×224 size, and encode them into +1280-dimensional features via MobileNetV2 network [68] +pre-trained on ImageNet [69]. Superpixel-level features are +computed as the mean of the patch-level features. Next, we +compute spatial features for each node by normalizing the +superpixel centroids by the image dimensions. Normaliza- +tion ensures the invariability of the spatial features to the +varying dimensions of input WSIs. Finally, we define the +TG edges by constructing a region adjacency graph topol- +ogy [70] using the spatial connectivity of superpixels. To +this end, we assume that adjacent tissue regions biologically +interact the most, and thus should be connected in a TG. +C. Contextualization of node embeddings +Given a TG, we learn discriminative node embeddings +(see Figure 1(b)) by using the node context information, +i.e., the tissue microenvironment and the inter-tissue inter- +actions. Specifically, we use GIN [59] denoted as Fθ. Since +GNNs can operate on graphs of arbitrary and varying sizes, +they allow to encode histopathology images represented in +form of TGs without needing tile-based processing. As the +discriminative information of a node relies on its local sub- +graph structures and can lie at different abstraction levels in +the GNN, we employ a Jumping Knowledge [71] strategy +to utilize multi-level node representations. Namely, the final +node-level embedding after T GIN-layers is defined as, +h(T )(v) = CONCAT(h(t)(v), ∀t ∈ {1, ..., T}) +(2) +where CONCAT denotes a concatenation operation. +D. WSI classification +Following the contextualization of node features, a graph- +classification head classifies the TG by using graph-level +embeddings hG and graph/image-level supervision. To ob- +tain a fix-sized hG, we use a READOUT operation that +averages the node embeddings h(T )(v), ∀v ∈ VG. Subse- +quently, hG is input to a multi-task classifier for primary +and secondary Gleason grading. Specifically, the classifier +includes two Multi-Layer Perceptrons (MLPs), denoted as +Fφ = {Fφ1, Fφ2}, to individually predict the primary, i.e., +the worst Gleason pattern, and secondary, i.e., the second- +worst Gleason pattern, in the WSI. Each MLP solves a multi- +class problem with |K| Gleason patterns, i.e., benign, Grade +3, Grade 4, and Grade 5. The final Gleason grade is derived +as the sum of the predicted primary and secondary patterns. +Fθ and Fφ are optimized jointly by minimizing the weighted +cross-entropy loss, +LG = λLCE(yGP , ˆyGP ) + (1 − λ)LCE(yGS, ˆyGS) +(3) +where, P and S denote primary and secondary labels of +ground truth yG and prediction ˆyG, and λ ∈ [0, 1] is a hyper- +parameter balancing the two terms. Further, during training +we introduce class-weights as w := {log( +� +i Ni +Ni +), +i = +{1, ..., |K|}}, where Ni is the count of class-wise Gleason +patterns in the training WSIs. These weights take care of the +class imbalance in Gleason grading by assigning a higher +value to classes with lower frequency. +E. Weakly supervised semantic segmentation +Nodes in a TG identify superpixels, i.e., morphologically +homogeneous tissue regions. Since each Gleason pattern is +characterized by distinct morphological patterns, we assume +that each tissue region, depicted by a node, includes a +unique Gleason pattern. Thus, the WSI segmentation task is +translated into classifying the nodes of the TG. In presence +of only image supervision, the node classification is achieved +in two steps. First, pseudo-node labels are generated by using +the image labels, and then, the pseudo labels are used to train +a node classifier. +Pseudo node labels: Following WSI classification, a post- +hoc feature attribution technique is used to measure the +importance of each node towards TG classification. Specif- +ically, we use GRAPHGRAD-CAM [72], [73], an exten- +sion of GRAD-CAM [74] to operate with GNNs. Given +a graph G, GRAPHGRAD-CAM produces class-wise node +attribution maps, Ak, ∀k ∈ K. These maps highlight the +importance ∀v ∈ VG for classifying G into |K|, as shown in +Figure 1. Provided the importance scores for v towards |K|, +it can be assumed that the label of v is k ∈ K, if the highest +importance score corresponds to class k. At this stage, an +argmax operation across Ak, ∀k ∈ K can be considered +to classify the nodes. However, such node labeling may be +suboptimal, because, +• Some nodes marginally contribute and bear low impor- +tance scores ∀k ∈ K for classifying a graph. However, +an argmax across the importance scores for a node +5 + +greedily selects the class with the highest score, even +though the node label is not ascertained. +• A node highly contributing towards the prediction of a +class is not necessarily part of this class. For example, +a node can bear high importance if it provides useful +complementary information for tie-breaking or ruling +out another class possibility. Formally, if the set of +nodes Vk ⊂ V has high importance scores for class +k, the labels of Vk are not ensured to be k. Even, the +labels of v ∈ Vk are not guaranteed to be the same. +• A class attribution map does not necessarily highlight +all the nodes belonging to the class. Depending on the +task complexity, a classifier may utilize only a subset of +the informative nodes from a class to predict the graph +label. Formally, if the set of nodes Vk ⊂ V have high +importance scores for class k, then Vk may not include +all the nodes in Vk ⊂ V that have the actual label k, +i.e., Vk ⊂ Vk. +• In presence of several feature attribution techniques in +literature, with different underlying mechanisms, can +produce different attribution maps [73]. Thus, a single +attribution technique may not be trusted for score-based +node classification. +We, therefore, strategize to use the highlighted nodes by +feature attribution as pseudo-labels to train a node-classifier. +For a graph G with Gleason score P+S, P, S ∈ K, we +compute node importance scores IP and IS, ∀v ∈ VG using +GRAPHGRAD-CAM. As the scores by GRAPHGRAD-CAM +are unbounded, we normalize the scores using min-max. +Then, we select the top n% nodes above a threshold t, +denoted as VP and VS, where n and t are hyperparameters +tuned during training. It selects the most informative nodes +for downstream node classification. For a node v ∈ VP +and v ∈ VS, we use arg max(IP (v), IS(v)) to ensure +VP ∩ VS = ∅. Finally, classes with the highest scores are +assigned as pseudo labels y ˜V to the nodes. Pursuing the +process for all the TGs in the dataset renders pseudo labels +Y ˜V . +Node classification: Y ˜V +is used to train the node- +classification head, as shown in Figure 1. Specifically for +a graph G, we get the node embeddings h(T )(v), ∀v ∈ VG +using Fˆθ, where ˆθ are the parameters of the GNN. Fˆθ +is frozen during node classification such that the same +GNN backbone is used for both segmentation and classifica- +tion, thereby reducing the number of trainable parameters. +h(T )(v), ∀v are processed by an MLP Fψ to predict Y ˜V . +Fψ is trained by optimizing a weighted multi-class cross- +entropy objective. Similar to the graph classification, class- +weights are defined as w := {log( +� +i Ni +Ni +), i = {1, ..., |K|}}, +where Ni is the number of annotated nodes of class i. The +node-wise predicted class labels are used to obtain the final +segmentation prediction. Noticeably, WHOLESIGHT does +not include any customized post-processing, unlike [26], +thus being applicable to various tissues and segmentation +tasks. +Notably, the graph- and the node-classification heads +address complementary tasks for a graph G, i.e., at graph- +level and at node-level, respectively. Therefore, following +the training of Fψ, we unfreeze Fθ, and jointly fine-tune ˆθ +and ˆψ with a small learning rate. The complementarity of +the tasks provides an additional informative signal to further +improve the segmentation and classification performance of +WHOLESIGHT. +IV. EXPERIMENTS +A. Datasets +We evaluate our method on three datasets containing +whole-slide prostate cancer needle biopsies for Gleason +pattern segmentation and Gleason grading. Gleason patterns +include grade 3 (G3)- moderately differentiated nuclei and +poorly-formed cribriform glands, grade 4 (G4)- poorly dif- +ferentiated nuclei and irregular masses, and grade 5 (G5)- +less differentiated nuclei and lack or only occasional glands. +Normal glands and non-epithelial tissues are labeled as +benign (B). Gleason grade depicts the worst (primary, P) +and the second-worst (secondary, S) Gleason patterns in a +WSI. Dataset details are as follows: +Sicap dataset: The dataset [75] contains 18,783 patches +of size 512×512 with complete pixel-level annotations and +slide-level Gleason grades for 155 WSIs from 95 patients. +The original slides and masks were reconstructed by stitch- +ing the patches. The WSIs were scanned at 40× magnifi- +cation by Ventana iS-can Coreo scanner and downsampled +to 10× magnification. The slides were annotated by expert +urogenital pathologists at the Hospital Cl´ınico of Valencia, +Spain. +Radboud dataset: [76] includes 5,759 needle biopsies +from 1,243 patients at the Radboud University Medical +Center, Netherlands. The slides were scanned with a 3D His- +tech Panoramic Flash II 250 scanner at 20× magnification +(resolution 0.24µm/pixel) and were downsampled to 10×. +Annotations include WSI Gleason grades and noisy pixel- +level Gleason pattern masks, released as part of the Prostate +cANcer graDe Assessment (PANDA) challenge [77]. The +masks were cleaned for segmentation using standard image +manipulation techniques, i.e., contextualized noise removal, +hole filling, and edge smoothing. In absence of large public +datasets with pixel-level annotated prostate cancer WSIs, we +used this dataset for developing and evaluating our method. +Karolinska dataset: The dataset [78] comprises of 5,662 +core needle biopsies from 1,222 patients at various hospitals +in Stockholm, Sweden. The slides were scanned with a +Hamamatsu C9600-12 and an Aperio Scan Scope AT2 +scanner at 20× magnification with a pixel resolution of +0.45202µm and 0.5032µm, respectively. The biopsies were +annotated by an expert uro-pathologist for Gleason grading. +6 + +Figure 2: Gleason grade-wise data distribution across train, validation, and test in Karolinska, Radboud and Sicap datasets. +Each dataset is split into train, validation, and test in +a ratio of 60%, 20%, and 20% at Gleason grade level, +using a random stratification that preserves the percentage +of classes in each split. The dataset distributions and splits +are displayed in Figure 2, which highlights the class-level +imbalances. +B. Implementation and evaluation +We implemented WHOLESIGHT using PyTorch [79], +DGL [80], and Histocartography [81], and conducted exper- +iments on NVIDIA Tesla P100 GPU and POWER9 CPU. +To develop the WHOLESIGHT network, Fθ, Fφ, and +Fψ were designed by optimizing their respective hyperpa- +rameters. First, Fθ and Fφ were trained by using image- +level labels, and then pseudo-node labels were created to +train Fψ to produce segmentation output. The number of +GIN layers in Fθ were optimized for the values {3, 4, 5}, +where the UPDATE function was defined as a 2-layer MLP +with 64 hidden units and ReLU activations. Fφ contains two +heads for classifying primary and secondary Gleason grades, +where each head consists of a 2-layer MLP with 128 hidden +units and ReLU activations. Fψ contains a 2-layer MLP with +128 hidden units and ReLU activations. +Considering the small size of the Sicap dataset, node- +level augmentations were employed to augment the training +dataset. Specifically, random node rotations {90, 180, 270} +degrees, and horizontal and vertical mirroring were used +for augmenting the nodes. Batch size and learning rate +were optimized from {4, 8, 16} and {10−4, 5×10−4, 10−3}, +respectively. Dropout layers with rates 0.2, 0.5 and 0.5 +were included in the MLPs belonging to Fθ, Fφ, and +Fψ, respectively. The pseudo-node labels were extracted +for selection percentages in {5, 10, 15, 20} and thresholds +{0.5, 0.6, 0.7}. Following the hyperparameter tuning, ten +WHOLESIGHT models were trained with different network +initializations. Validation weighted-F1 was used for model +selection. The reported results correspond to the mean and +standard deviation over these ten models. +Classification metrics: Classification performance was +measured by the weighted-F1 score of Gleason grade and +the quadratic kappa score (κ2) of ISUP grade [82], [83]. +ISUP is an alternate grading system whose correspondence +with Gleason grading is defined as, Benign → ISUP-0, GG- +(3+3) → ISUP-1, GG-(3+4) → ISUP-2, GG-(4+3) → ISUP- +3, GG-8 → ISUP-4, and GG≥9 → ISUP-5. κ2 captures the +degree of disagreement between the prediction and ground +truth labels. For example, a grade 6 sample predicted as +grade 10 is penalized more than predicting grade 7. +Segmentation metrics: Segmentation performance was +measured by Dice score. Given the imbalance of the Gleason +patterns in the datasets, we also reported the per-pattern Dice +score. +Uncertainty metrics: Following the work of [84], we +evaluated the classification uncertainties in terms of Brier +score sB (lower is better) and the NLL sNLL (lower is better) +over a set of N test samples, defined as, +sB = 1 +N +N +� +n=1 +|K| +� +i=1 +(yi − ˆyi)2, +sNLL = − 1 +N +N +� +n=1 +|K| +� +i=1 +p(yi) log ˆp(yi) +(4) +Calibration metrics: Reliability diagrams provide an intu- +itive understanding of model calibration. To quantify these +observations, we used the Expected Calibration Error (ECE) +metric [85], which computes the weighted average deviation +of the confidence scores over all the bins, i.e., +cECE = +B +� +b=1 +Nb +N |acc(b) − conf(b)| +(5) +where nb is the number of samples in bin b, acc(b) and +conf(b) are the accuracy and the average confidence of +samples in b. +C. Baselines +We compared WHOLESIGHT with state-of-the-art WSI +classification methods and two variants of WHOLESIGHT. +7 + +Sicap +Radboud +Karolinska +Train + Train +Train +0.40 +0.40 +0.40 - +Val +Val +Val +1920 +0.35 - +Test +0.35 +Test +0.35 +Test +1812 +1600 +0.30 +0.30 +0.30 - +8 +36 +36 +31 +29 +963 +985 +852 +860 +% +% +% +764 +0.15 - +0.15 +0.15 - +16 +0.10 +479 +0.10 - +0.10 +235 +0.05 - +0.05 - +0.05 +109 +16 +0.00 +0.00 +0.00 - +Benign GG6 +Benign GG6 +GG7 +GG7 +GG8 +GG9 GG10 +GG7 +GG8 +GG9GG10 +Benign GG6 +GG8 +GG9 +GG10FSConv: We implemented the two-step method proposed +by [75] for WSI classification. First, we extracted patches +of size 256×256 from WSIs and classified them using +FSConv+global-max pooling. The patches were labeled us- +ing the Gleason pattern masks, and patches with >90% +homogeneous pattern were selected for classifier training. +During inference, dense patch predictions produced the +output segmentation masks. An MLP was trained on the +Gleason grade percentages over the WSI patches for Gleason +grading. +WHOLESIGHT(Graph, GRAPHGRAD-CAM): In com- +parison to WHOLESIGHT, this baseline contained only Fθ +and Fφ. It did not create or utilize pseudo labels, and the +segmentation output was obtained by taking the argmax over +the class-wise GRAPHGRAD-CAM attribution maps. +WHOLESIGHT(Multiplex, NC): This variant used both +image- and pixel-level supervision during training and acts +as the upper bound for WHOLESIGHT. As pixel-level +annotations were available, Fψ was trained using ground- +truth node-level labels, instead of generating pseudo-node +labels. The model consisted of the same Fθ, Fφ, and Fψ +as the WHOLESIGHT architecture. In this baseline, Fθ, +Fφ, and Fψ were trained jointly by optimizing a multi- +task objective, i.e., WSI-level primary and secondary Glea- +son score prediction along with node-level Gleason pattern +prediction. This variant of WHOLESIGHT was proposed in +our preliminary work, as described in [29]. +Multiple Instance Learning (MIL): +MIL +methods +are +state-of-the-art for WSI classification. In particular, we com- +pared to ABMIL [30], which used an attention mechanism to +aggregate patch embeddings into a fix-sized WSI embedding +that was fed to a classifier for Gleason grading. We also in- +cluded CLAM [14], a method built on ABMIL by including +an additional constrain to cluster similar patch embeddings. +Our experiments followed the public implementations * with +adjustments to enable multi-task classification. +For all the baselines, hyper-parameters are thoroughly +tuned to use the best learning rate and batch size, if +applicable. Subsequently, ten models were re-trained from +scratch with the optimal parameters. We report the mean +and standard deviation over these runs for each experiment. +D. WSSS performance analysis +We studies the classification and segmentation perfor- +mance of WHOLESIGHT and the competing methods by +independently training and testing them on Sicap, Radboud, +and Karolinska. +Performance analysis: Table I presents the results on +Sicap. The analyses are grouped into two supervision set- +tings, i.e., complete (C) and weak (W). Setting-C utilizes +both image- and pixel-level annotations, whereas, Setting- +W only uses image-level labels. WHOLESIGHT reached +*CLAM publicly available code: https://github.com/mahmoodlab/CLAM +37.6% average Dice score, which significantly outperforms +WHOLESIGHT (Graph, GRAPHGRAD-CAM) by +6.6% in +absolute. WHOLESIGHT (Multiplex, NC), that acts as the +upper bound, produced a significant gain in segmentation +compared to WHOLESIGHT. The per-class Dice scores +indicate that the benign patterns that constitute most tissue +areas have a high detection rate compared to less occurring +Gleason patterns. For the classification task, WHOLESIGHT +outperformed ABMIL and CLAM, both in terms of Gleason +grade weighted-F1 and ISUP κ2. Notably, +Table II presents the results on Radboud. WHOLESIGHT +rendered an absolute gain of +4.8% in average Dice score +over WHOLESIGHT (Graph, GRAPHGRAD-CAM). This +confirms the utility of pseudo-node labels for superior +segmentation. Similar to the observations on Sicap, benign +patterns had a high detection rate, followed by G3, G4, and +G5 patterns. As Radboud dataset includes more G5 patterns +than Sicap, we observed a significant gain in detecting +high-grade Gleason patterns. For the classification task, the +observations were also consistent with the observations on +Sicap. +Table III presents the results on Karolinska. In absence of +ground truth pixel-level annotations, the segmentation per- +formances could not be computed. WHOLESIGHT (Graph) +outperformed the baselines in terms of classification perfor- +mance. +The observations across Table I, III and III conclude that, +jointly optimizing classification and segmentation objectives +provide complementary information to improve the overall +classification performance, i.e., WHOLESIGHT (Multiplex) +> WHOLESIGHT > WHOLESIGHT (Graph). +E. Generalization: performance, uncertainty, and calibra- +tion +We studied the generalization ability of WHOLESIGHT +following a modified training setting. Specifically, we used +Radboud and Karolinska training WSIs for model training. +Thus, the training set encompassed better sample variability +and diagnostically more challenging cases than the stan- +dalone training counterparts on individual datasets. Testing +was performed individually on Radboud and Karolinska test +WSIs, herein studying the in-domain performance. Further, +we tested on the entire Sicap dataset, which consisted of +out-of-domain WSIs. +Performance analysis: Table IV compared the classifi- +cation performance of WHOLESIGHT and the competing +baselines. In terms of the weighted-F1 score on the in- +domain test set, WHOLESIGHT outperformed ABMIL and +performed better or comparable to CLAM. Similar patterns +were also observed for ISUP κ2. However, the variances of +the classification for WHOLESIGHT is consistently lower +than ABMIL and CLAM. When tested on the out-of-domain +Sicap dataset, WHOLESIGHT achieved significantly better +classification than ABMIL and CLAM. +8 + +Table I: Classification and segmentation results on Sicap dataset. The best performances for using image-level supervision +are highlighted in bold. +Annot. +per-class Dice +avg. Dice GG wF1 ISUP κ2 +Method +Benign +Grade3 +Grade4 +Grade5 +C +FSConv [75] +65.7±0.5 24.4±1.4 29.0±1.4 8.4±0.6 +31.9±0.5 +48.7±3.4 50.9±3.4 +WHOLESIGHT (Multiplex, NC) +92.5±0.3 35.4±2.3 51.6±2.0 11.1±2.2 47.6±2.1 +59.6±4.1 84.6±3.2 +W +ABMIL [30] +- +- +- +- +- +50.2±6.3 67.8±5.2 +CLAM [14] +- +- +- +- +- +51.4±5.5 75.2±4.7 +WHOLESIGHT +64.4±6.1 23.1±2.0 32.8±6.9 3.7±1.0 +31.0±2.7 +53.3±5.3 81.9±6.7 +(Graph, GRAPHGRAD-CAM) +WHOLESIGHT +67.6±0.5 28.6±0.3 49.1±0.4 5.0±0.4 +37.6±0.3 +58.6±6.2 89.0±1.2 +(Graph + Pseudo nodes, Node class.) +Table II: Classification and segmentation results on Radboud dataset. The best performances for using image-level +supervision are highlighted in bold. +Annot. +per-class Dice +avg. Dice GG wF1 ISUP κ2 +Method +Benign +Grade3 +Grade4 +Grade5 +C +FSConv [75] +84.3±0.1 53.3±0.5 62.5±0.3 36.9±0.5 59.2±0.1 +45.9±1.3 69.4±0.6 +WHOLESIGHT (Multiplex, NC) +91.5±0.1 63.9±0.4 66.2±0.3 36.8±1.2 64.6±0.4 +68.9±0.9 83.8±0.9 +W +ABMIL [30] +- +- +- +- +- +59.3±1.7 79.7±1.2 +CLAM [14] +- +- +- +- +- +60.3±1.6 80.2±1.5 +WHOLESIGHT +71.3±2.2 26.9±1.0 24.9±1.2 15.1±0.5 34.6±0.6 +66.0±1.0 82.2±0.5 +(Graph, GRAPHGRAD-CAM) +WHOLESIGHT +75.9±0.3 32.9±1.0 29.1±1.5 19.6±0.7 39.4±0.3 +67.9±0.3 83.0±0.2 +(Graph + Pseudo nodes, Node class.) +Table III: Classification results on Karolinska dataset. The best performances for using image-level supervision are +highlighted in bold. +GG wF1 ISUP κ2 +W +ABMIL [30] +65.0±2.0 79.1±1.2 +CLAM [14] +63.6±2.5 77.6±2.0 +WHOLESIGHT (Graph) 70.5±0.6 80.2±0.7 +Table IV: Classification and segmentation results on Radboud, Karolinska, and Sicap datasets for models trained using both +Radboud and Karolinska datasets. +Annot. +Radboud +Karolinska +Sicap +Method +avg. Dice GG wF1 ISUP κ2 GG wF1 ISUP κ2 avg. Dice GG wF1 ISUP κ2 +C +FSConv [75] +59.2±0.1 +45.9±1.3 69.4±0.6 34.5±1.1 40.1±1.3 49.5±0.4 +52.1±2.1 53.8±1.7 +WHOLESIGHT (Multiplex, NC) +64.5±0.3 +69.0±1.0 83.6±0.9 71.2±0.7 82.5±1.3 60.0±0.5 +65.5±2.5 85.6±2.8 +W +ABMIL [30] +- +57.6±2.3 73.8±2.3 65.5±1.3 77.3±2.8 - +56.4±2.7 75.0±7.5 +CLAM [14] +- +61.7±2.1 78.6±1.3 69.3±1.3 82.8±1.0 - +53.1±3.8 74.6±4.2 +WHOLESIGHT +34.6±0.6 +66.0±1.0 82.2±0.5 69.2±0.9 80.3±0.9 30.4±1.0 +65.1±2.3 86.1±2.5 +(Graph, GRAD-CAM) +WHOLESIGHT +44.6±0.2 +66.2±0.1 82.9±0.1 70.2±0.1 81.3±0.1 42.0±0.3 +65.2±0.1 86.6±0.1 +(Graph + Pseudo nodes, Node class.) +Confusion +matrices +for +the +best +Gleason +grading, +ISUP grading, primary- and secondary classification with +WHOLESIGHT are presented in Figure 3 on the three +datasets. It can be observed that most misclassifications lie +close to the diagonal. Majority of the confusion occurred +between GG6 and GG7, i.e., GG(3 + 3) versus GG(3 + +4) and GG(4 + 3). Such ambiguity is prevalent among +pathologists, as shown in [86], [87]. High-grade Gleason +grading better on Radboud than Karolinska due to more +number of high-grade samples in Radboud. Primary- and +secondary classification weighted-F1 for Radboud, Karolin- +ska and Sicap were 79.3%, 81.7%, 81.6% and 62.5%, +69.7%, 64.5%, respectively. This indicated that identifying +secondary Gleason pattern is more challenging. Table IV +9 + +Figure 3: Confusion matrices for Gleason grading, ISUP grading, primary- and secondary Gleason classification on Radboud, +Karolinska, and Sicap datasets for with the best WHOLESIGHT model trained using Radboud and Karolinska training +datasets. +also presents the generalizability assessment of segmen- +tation. WHOLESIGHT consistently performed better than +WHOLESIGHT (Graph, GRAPHGRAD-CAM). Noticeably, +the Dice scores on Radboud and Sicap datasets improved +over the segmentation results in Table II and I by 5.2% and +4.4% for WHOLESIGHT. It can be reasoned to the usage +of more training WSIs, which indicate that WSSS can be +improved by utilizing more weak supervision. +Uncertainty analysis: Figure 4 presents the classification +uncertainty analysis of WHOLESIGHT, WHOLESIGHT +(Graph, GRAD-CAM), and WHOLESIGHT (Multiplex, +NC), in terms of NLL and Brier score, on Radboud and +Karolinska datasets. WHOLESIGHT (Multiplex, NC) ren- +dered a significantly lower NLL than WHOLESIGHT across +all datasets for primary, secondary, and Gleason grade (P+S) +classification. Noticeably, the NLL and Brier scores were +consistently higher for predicting the secondary Gleason +patterns than the primary patterns. This resonates with the +fact that identifying secondary patterns is more challenging +with higher ambiguity. +Model calibration analysis: A model with good uncer- +tainty estimate should be well-calibrated, i.e., the model con- +fidence should be close to the model performance. Figure 4 +presents the reliability diagrams of the primary classification +head on Karolinska and Radboud datasets. WHOLESIGHT +showed consistently better calibration than WHOLESIGHT +(Graph, GRAD-CAM) and similar calibration with respect +to WHOLESIGHT (Multiplex, NC). ECE also metric quan- +titatively supported this observation. However, we observed +that still all models remains over-confident as the model +accuracies over the confidence bins remained lower than the +expected calibration (in blue). +10 + +Gleason grade +ISUP grade +Primary Gleason + Secondary Gleason +Benign +7 +2 +5 +3 +0 +Benign +179 +7 +0 +2 +5 +3 +Benign +179 +7 +10 +0 +Benign +179 +9 +5 +3 +Grade6 +89 +21 +17 +3 +0 +0 +Gradel +17 +15 +6 +0 +3 +Radboud +Grade3 - +185 +45 +label +21 +23 +197 +56 +26 + label +1 +True label +Grade7 +192 +24 +22 +Grade2 +28 +52 +31 +0 +10 +35 +4 +3 +True +Grade8 +24 +56 +24 +Grade3 - +7 +16 +3 +93 +21 +23 +4 +6 +14 +20 +Grade4- +31 +330 + Grade4- +82 +121 +55 +9 +Grade9 . +22 +35 +75 +22 +Grade4 - +1 +3 +5 +19 +56 +32 +5 +4 +32 +28 +Grade5. +Grade5 +2 +1 +21 +41 +74 +5 +0 +1 +22 +38 +109 +Gradel0 - +3 +4 +8 +Grade5 - +6 +1 +1 +ex +ade2 +ade4 +Gra +Gr +Gr +Gr +Gr +Predicted label +Predicted label +Predicted label +Predicted label +0 +Benign +30 +2 +0 +Benign +2 +2 +343 +30 +2 +343 +2 +32 +0 +Benign- +343 +Benign +343 +30 +4 +2 +4 +Gradel +0 +Grade6 . +253 +30 +0 +0 +253 +30 +60 +6 +0 +6 +60 +Karolinska +70 +378 +26 + Grade3. +63 +274 +69 +2 +True label +1 +Grade7 . +59 +86 +21 +True label +Grade2 - +56 +39 +13 +2 +1 +10 +8 +9 +rue +Grade8 - +27 +51 +Grade3 - +5 +3 +21 +18 +12 +6 +2 +1 +3 +12 +40 + Grade4 +17 +107 +10 +138 +5 +89 +Grade9 +0 +8 +16 +15 +Grade4 +7 +7 +20 +6 +6 +2 +2 +6 +19 +Grade5- +0 +6 +Grade5- +2 +20 +0 +Grade10 +0 +0 +0 +0 +2 +Grade5 +3 +5 +16 +0 +0 +24 +2 +Gr +Predicted label + Predicted label +Predicted label +Predicted label +Benign +Benign +34 +1 +0 +1 +0 +0 +34 +0 +0 +0 +1 +Benign +Benign. +34 +1 +0 +34 +1 +1 +0 +1 +0 +Gradel +Grade6 . +6 +0 +0 +24 +1 +0 +0 +5 +2 +Grade3 - +33 +16 + Grade3 - +37 +0 +2 +True label +Grade7 . +19 +4 +label +Grade2 +3 +5 +5 +1 +0 +0 +1 +0 +0 +0 +Sicap +True +Grade8 +10 +15 +8 +Grade3 +1 +0 +2 +7 +6 +1 +12 +54 +10 +14 +22 +15 +5 +8 +1 +Grade4 - +1 +９ +9 +Grade9 . +0 +1 +1 +1 +Grade5. +1 +0 +7 +3 +Grade5 +2 +6 +11v +1 +0 +2 +2 +2 +1 +0 +1 +7 +13 +Gradel0 - +0 +Grade5 - +1 +1 +e +de4 +ade5 +ade2 +e +Predicted label +Predicted label +Predicted label +Predicted labelFigure 4: Uncertainty and model calibration analysis of WHOLESIGHT, WHOLESIGHT (Graph, GRAD-CAM), and +WHOLESIGHT (Multiplex, NC) models for Radboud (a, b, c) and Karolinska (d, e, f) datasets. (a, d) and (b, e) present NLL +(lower is better) and Brier scores (lower is better), respectively. (c, f) present reliability diagrams of the primary Gleason +classification head. Expected calibration (blue) highlights a perfectly calibrated model. Calibrations of WHOLESIGHT, +WHOLESIGHT (Graph, GRAD-CAM), and WHOLESIGHT (Multiplex, NC) are in orange, red, and purple, respectively, +along with the number of samples (in %) in each bin. +F. Qualitative analysis +We qualitatively analyze the results of WHOLESIGHT by +(1) visualizing overlaid segmentation masks on WSIs, (2) +analyzing the t-distributed stochastic neighbor (t-SNE) [88] +node embeddings, and (3) correlating the segmentation out- +puts with pathological reasonings. +Segmentation mask visualization: Figure 5 demonstrates +segmentation predictions obtained with WHOLESIGHT and +WHOLESIGHT(Multiplex, NC) on Sicap dataset. We can +11 + +(a) +Radboud +(d) +Karolinska +P+$ +P +s +P+$ +P +s +2.0 - +2.0 - +T +1.5 +1.5 +T +TTN +TIN +1.0 +1.0 - +0.5 +0.5 +(b) +(e) +P+S +P +P+S +s +P +s +0.7 +0.7 +0.6. +0.6 +Brier score +Brier score +0.5 +0.5 +0.4 +0.4 +0.3 - +0.3. +0.2 +0.2 +(c) +() +100 +-100 +100 +卜100 +-○-- Expected calibration +--. Expected calibration +-○-. WholeSIGHT Graph- ECE=0.27 +-○- WholeSIGHT Graph- ECE=0.27 +-○-- WholeSIGHT- ECE=0.23 +-○-. WholeSIGHT- ECE=0.21 +80 - +:-○-- WholeSIGHT Multiplex- ECE=0.22 +80 +.-○-. WholeSIGHT Multiplex- ECE=0.23 +80 +Accuracy (%) +60 +60 +60 +.40 +40 / +40 +40 +20 +20 +20 +20 +T0 +0 +0 +50 +60 +70 +80 +90 +100 +50 +60 +70 +80 +90 +100 +Confidence (%) +Confidence (%)Figure 5: Sample segmentation maps from Sicap dataset. Ground truth is shown on the left, WHOLESIGHT predictions +in the middle, and WHOLESIGHT(Multiplex, NC) on the right. Tissue regions, i.e., TG nodes, are represented by black +overlay. (a, b, c) display GG(3+3), GG(4+4), and GG(5+5) samples, respectively. For better visualization, benign areas are +not highlighted in the segmentation maps. +12 + +(a) +Gleason grade 3+3 +Benign +G3 +CA +b +Gleason grade 4+4 +Benign +G3 +G4 +G5 +(c) +Gleason grade 5+5 +Benign +G3 +G5 +Ground truth +WholeSIGHT +WholeSIGHT (Multiplex, NCFigure 6: t-SNE visualization of tissue-graph node embeddings and example patches from several regions on the two- +dimensional t-SNE feature space for Sicap dataset. (a) t-SNE visualization of the correctly classified nodes. (b) and (c) +display the t-SNE visualization of misclassified nodes, where (b) and (c) highlight the ground truth and predicted node +labels, respectively. (d) and (e) demonstrate square patches of size 224×224 at 10× magnification cropped around the node +centroids selected from different regions on the t-SNE embedding space. (d) and (e) highlight the correctly and incorrectly +classified node patches, respectively. The labels of the patches in (e) are formatted as Y → ˆY , where Y and ˆY denote the +ground truth and the predicted class label. The colored rectangles around the patches in (d) and (e) correspond to respective +colored rectangles in (a), (b), and (c). +13 + +(a) +Benign +Gleason 3 +Gleason 4 +Gleason 5 +(d) +100 - +Correct classifications +50 +0 +G3 +-50 +G4 +-100 +-100 +-50 +0 +50 +100 +(b) +G5 +Benign +Gleason 3 +Gleason 4 +Gleason 5 +Misclassifications: Ground truth +(e) +100 +→G3 +50 +B +→ G4 +0 +B +G5 +50 +B +100 +B +100 +-50 +0 +50 +100 +() +Benign +Gleason 5 +G3 +Gleason 3 +Gleason 4 +G4→B +Misclassifications: Prediction +100 +50 +→ G3 +G4 +0 +→ G5 +50 +G4 +100 +G5 +100 +-50 +0 +50 +100observe that WHOLESIGHT correctly delineates the can- +cerous regions in the WSIs. Zooming into different regions +conclude that the tissue regions of TG, i.e., the nodes of +TG, (outlined in black in Figure 5) encode meaningful units +of homogeneous tissue. It substantiates the relevance of +using TG representations for segmenting tissue regions into +Gleason patterns. We further notice that WHOLESIGHT, in +a few cases, predicts benign regions adjacent to cancerous +patterns as cancerous. For example, the benign region, +primarily consisting of stroma, in Figure 5(c) is predicted +as G5. We argue that these false positive detections do +not inhibit the applicability of the method, as neighboring +cancerous regions are correctly detected. In a few other +cases, WHOLESIGHT correctly detects missed cancerous +regions in the ground truth annotations. For instance, in +Figure 5(b), the missing G4 region in the upper part of the +WSI is correctly identified. +Comparing WHOLESIGHT with WHOLESIGHT (Mul- +tiplex, NC), we observe that several false positives are re- +moved, e.g., in Figure 5(a), thus offering more accurate seg- +mentation. However, the improvements by WHOLESIGHT +(Multiplex, NC) are achieved at the cost of training with +pixel-level annotations that are hardly available in real- +world practice. Thus, WHOLESIGHT appears to be an +appealing compromise between segmentation performance +and annotation requirement. +Visualizing t-SNE feature space: A t-SNE visualization of +the learned tissue-level embeddings is demonstrated in Fig- +ure 6 for Sicap. t-SNE projects the GNN node embeddings +onto a two-dimensional feature space, allowing to analyze +the connection between node embeddings and the Gleason +pattern distribution. +Figure 6(a) displays the t-SNE feature space for the cor- +rectly classified nodes, which highlights demarcated clusters +for each Gleason pattern. The large cluster of benign nodes +indicates the variability of the benign tissue. Several patches +from each Gleason pattern cluster are presented in Fig- +ure 6(d). We can observe the reduced nuclei differentiation +across the patches from benign to Gleason grade 5. Further, +Figure 6(b) and (c) display the t-SNE feature space for the +misclassified nodes. Specifically, Figure 6(b) presents the +ground truth node labels, and Figure 6(c) the predicted node +labels. Different embedding locations are further selected +and highlighted by different colored rectangles and put in re- +lation with corresponding patches to indicate the inter-class +ambiguities, as demonstrated in Figure 6(e). For example, +the first row in Figure 6(e) showcases patches that are benign +but are predicted as G3. We can visually compare these +patches with the G3 patches in the third row of Figure 6(d). +Similar ambiguities between other pairs of Gleason patterns +are also included in Figure 6(e). +Interpreting model outcomes via predicted segmentations: +Predicted +segmentations +provide +human-understandable +interpretability maps. For researchers, the segmentations +allow to, (1) identify morphological patterns responsible for +WSI classification, (2) analyze failure cases by inspecting +pixel-level predictions, and ultimately (3) better understand +the model behavior towards biomarker discovery. For +pathologists, they assist to, (1) put in relation the predicted +WSI-level Gleason scores and the highlighted pixel-level +Gleason patterns, (2) confirm that the morphology of the +identified cancerous regions aligns with pre-established +diagnostic criteria. +Additionally, in the perspective of developing AI-assisted +human-in-the-loop tools, a Gleason grading system that can +simultaneously classify and segment WSIs is closer to the +latest pathological standards. Indeed, recent revisions of the +Gleason grading system [83] emphasized the importance +of reporting the percentage of each grade for better pa- +tient stratification and treatment selection [89]–[92]. These +percentages can be trivially derived from the predicted +segmentation maps by counting the number of pixels be- +longing to each pattern. Naturally, such information is not +available in mere WSI classification systems. Reporting per- +grade percentage is particularly important in ambiguous and +borderline cases. For instance, consider two patients with +Gleason score 3+4. When a small percentage of pattern- +4 is present, e.g., 10%, the case can be considered as an +intermediate risk cancer where active patient surveillance +is enough [93]. However, a larger secondary pattern may +require specific treatments. Reporting percentages of each +grade allows us to discriminate between these two scenarios +easily. +Similarly, consider a Gleason score 4+3 with a small +secondary Gleason pattern, e.g., 90% and 10% area for +primary and secondary patterns, respectively. This case will +be scored as 4+3, even though it is close to a score of +4+4, which would lead to a different treatment protocol. By +explicitly reporting the Gleason pattern percentages, such +corner cases can be avoided. +V. CONCLUSION +Accurate +delineation +of +patterns +in +whole-slide +histopathology +images +typically +demands +pixel-level +annotations, which are hard to acquire in a real-world +scenario. +Nonetheless, +the +semantic +segmentation +of +diagnostically +relevant +patterns +is +crucial +for +disease +diagnosis and treatment selection. To this end, we proposed +a novel weakly-supervised semantic segmentation method, +WHOLESIGHT, that can segment the relevant patterns of +interest in histopathology images by leveraging only image- +level supervision. To our knowledge, WHOLESIGHT is +the first weakly-supervised semantic segmentation method +that can operate in an end-to-end manner on histopathology +images of arbitrary shape and size. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Brigham and Women’s Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Harvard Medical School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' USA 3Department of Pathology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Massachusetts General Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Harvard Medical School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' USA 4Cancer Program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Broad Institute of Harvard and MIT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' USA 5Data Science Program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dana-Farber/Harvard Cancer Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' USA 6Computer-Assisted Applications in Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Switzerland 7Signal Processing Laboratory 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' EPFL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Switzerland 8EPFL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Switzerland 9Department of Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Uppsala University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Sweden Abstract—The segmentation and automatic identification of histological regions of diagnostic interest offer a valuable aid to pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, segmentation methods are hampered by the difficulty of obtaining pixel-level annotations, which are tedious and expensive to obtain for Whole-Slide images (WSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To remedy this, weakly supervised methods have been developed to exploit the annotations directly available at the image level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, to our knowledge, none of these techniques is adapted to deal with WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In this paper, we propose WHOLESIGHT, a weakly-supervised method, to simultaneously segment and classify WSIs of arbitrary shapes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, WHOLESIGHT first constructs a tissue-graph representation of the WSI, where the nodes and edges depict tissue regions and their interactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' During training, a graph classification head classifies the WSI and produces node-level pseudo labels via post-hoc feature attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' These pseudo labels are then used to train a node classification head for WSI segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' During testing, both heads simultaneously render class prediction and segmentation for an input WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We evaluated WHOLESIGHT on three public prostate cancer WSI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Our method achieved state-of-the-art weakly-supervised segmentation performance on all datasets while resulting in better or comparable classifi- cation with respect to state-of-the-art weakly-supervised WSI classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Additionally, we quantify the general- ization capability of our method in terms of segmentation and classification performance, uncertainty estimation, and model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Keywords-Computational Pathology, Whole-Slide image seg- mentation, Weakly supervised learning, Weakly supervised classification, Weakly supervised segmentation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' INTRODUCTION Prostate cancer is the second most frequently diagnosed cancer in men in the United States, with 250,000 new reg- istered cases resulting in 35,000 deaths in 2022 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Yet the Corresponding author: Pushpak Pati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Email: pus@zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='com number of pathologists, whose role is critical in the diagnosis and management of cancer patients, is gradually declining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In the United States, an 18% decrease was recorded between 2007 and 2017, resulting in a 42% increase in the average workload [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In addition, the practice of uro-pathology also has its share of challenges [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Although the diagnostic criteria for grading prostate cancer are established [4], the continuum of phenotypic features across the diagnostic spec- trum leaves room for disparities, with significant intra- and interobserver variability [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The manual inspection of slides is also tedious and time-consuming, and would benefit from automation and standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' These elements justify the development of Computer-Aided Diagnosis (CAD) tools to automate the diagnostic workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To this end, several Artificial Intelligence (AI) based CAD tools are proposed, including nucleus segmentation and clas- sification [7]–[9], gland segmentation [9]–[11], and tumor detection [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Albeit the remarkable performance, these tools often demand task- and tissue-specific annota- tions on large datasets, which are tedious, time-consuming and often infeasible to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For reducing annotation requirements, different approaches are proposed, in particu- lar, weakly-supervised methods based on Multiple Instance Learning (MIL) framework for the automatic classification of Whole-Slide Images (WSIs) [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Although classification is useful, it remains limited in its role of supporting the pathologist’s attention during diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In this context, semantic segmentation methods are preferable as they enable the generation of pixel-level delineation of the tissue constituents that can highlight diagnostically relevant regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Such visualization allows for strengthening trust between pathologists and CAD tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Additionally, the identified regions can be leveraged by a classifier to improve patient diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, semantic 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='02933v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='CV] 7 Jan 2023 segmentation generally requires pixel-level labels, which makes it more demanding in terms of annotations than classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For this reason, the development of weakly-supervised semantic segmentation (WSSS) methods appears as the most adequate response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' While WSSS has been successful on natural images, it en- counters various challenges when applied to histopathology images [16], as they, (1) contain fine-grained objects with large intra-class variations [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (2) often include ambiguous boundaries among histology components [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (3) can be several giga-pixels with arbitrary tissue sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Nevertheless, some WSSS methods are proposed for various histology tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The methods by [19]–[25] performing WSSS at patch- level are limited by the need for patch-level annotations, and inability to perform global contextualized WSI seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' While [26], [27] scale to larger tiles, they pose high computational complexity and memory requirements for operating on WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The methods by [25], [26] require exact tile annotations for model training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', a precise denomination of each lesion type in a tile, which requires pathologists to annotate images beyond standard clinical needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' On a different note, recent WSI classification methods use attention mechanisms or feature attributions to highlight salient regions [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Though these regions are informative for visual assessment, they are insufficient, incomplete, and blurry for accurately delineating relevant regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Addition- ally, producing granular saliency requires densely overlap- ping patch predictions, which is computationally expensive while working with WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In view of the aforementioned limitations of the WSSS methods, we propose WHOLESIGHT, “Whole- slide SegmentatIon using Graphs for HisTopathology”, that can simultaneously segment and classify arbitrar- ily large histopathology images by using WSI-level la- bels, and without any task-specific assumptions or post- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, WHOLESIGHT transforms an im- age into a superpixel-based tissue-graph (TG), and con- siders the segmentation problem as a node-classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT incorporates both local and global tissue microenvironment to perform contextualized segmen- tation, principally in agreement with inter-pixel relation- based WSSS [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To summarize, our contributions are: WHOLESIGHT, a novel graph-based weakly- supervised method to jointly segment and classify WSIs using readily available WSI-level annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' A comprehensive evaluation of WHOLESIGHT on 3 prostate cancer datasets for Gleason pattern segmen- tation and Gleason grading, and benchmarked against state-of-the-art WSI-level weakly-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Thorough generalizability quantification of WHOLESIGHT on in- and out-of-domain cohorts in terms of segmentation and classification performance, uncertainty estimation, and calibration of neural network predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' A preliminary version of this work was presented as [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Our substantial extensions herein include, (1) an improved WHOLESIGHT method in terms of model architecture and automatic synthesis of node labels, (2) extensive evaluations on large cohorts of WSIs (approximately 100×), and (3) generalization assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Weakly-supervised histopathology image classification Weakly-supervised classification of WSIs has been mostly developed around MIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In MIL, a WSI is first decomposed into a “bag” of patches and are encoded by a neural encoder, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', a Convolutional Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Then, an aggregator pools the patch embeddings to produce a slide- level representation for mapping to a class label via a neural predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The aggregator can be based on an attention mechanism weighing the importance of each patch, as in [14], [30], or as recently proposed, it can take the form of a transformer [31], [32] or a Graph Neural Network (GNN) [33], enabling modeling inter-patch dependencies and global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Differently, context can be modeled using multi-scale representations of WSIs, either via multi- magnification patch embeddings [23], [34] or by learning to automatically select important regions, as proposed in [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Despite the success of these approaches, they cannot directly be extended for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Weakly-supervised histopathology image segmentation WSSS approaches in histopathology can be categorized by the type of supervision (or annotation), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', point an- notations, scribbles, or image-level labels, and the scale of operation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', patches, tiles, Tissue-Micro Array (TMAs), or WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [37]–[39] utilized point annotations to segment cells and nuclei in histology patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [23], [40] used scribble annotations to segment tissue and tumor regions, respec- tively, at patch-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Both the approaches used U-Net [41], where [23] leveraged concentric patches across multiple magnifications for including relevant context information, and [40] modified the objective function to balance the contribution of the annotated pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The majority of the WSSS methods in histopathology utilized image-level su- pervision and are limited to operate with patch annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [19] proposed multiple clustered instance learning to process sliding patches for simultaneous grading and segmentation of colon TMAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [21] trained a binary classifier for pixel- level predictions and afterward computed an image-level prediction from pixel labels via a softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' They optimized image prediction, such that pixel predictions were improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [22] proposed CAMEL, a MIL-based label en- richment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It split an image into latticed instances, generated instance labels, and assigned instance labels to corresponding pixels to enable supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [26] proposed HistoSegNet, which trained a CNN to predict tissue types in a tile and used feature attribution to derive 2 pixel-level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It also employed a series of dedicated post-processing steps for prediction refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [24] used foreground proportion as the weak labels and combined a fully convolutional network and a graph convolutional network for tissue segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' [25] proposed a feature attribution-based model to generate pseudo labels, followed by a multi-layer pseudo-supervision network for segmenting tissue types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' As a main limitation, these methods cannot perform WSSS on WSIs using only WSI labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To perform WSSS beyond patch-level, [27] proposed WeGleNet, that scales to TMAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It included a segmentation- and a global- aggregation layer to classify images during training, and up- sampled pixel-level softmax activations during inference for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, the method cannot precisely delineate lesions and highlight multiple lesion occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It also requires processing densely overlapping patches for fine segmentation, and cannot scale to WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In contrast, our WHOLESIGHT can perform WSSS by leveraging image- level supervision, while efficiently scaling to WSIs of arbi- trary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Generalization quantification in histopathology Generalizability of CAD tools in histopathology is af- fected by domain-level biases, which are introduced due to numerous reasons, such as different staining protocols, manufacturing devices, materials, and scanning devices with respective color response [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Though generalizable tools, that are robust to domain shifts, are desired, it is challenging to model and detect the domain shifts in Deep Learning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Nevertheless, several approaches have been proposed to reduce such domain shifts via data- and model-level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Data-level adaptation can be achieved via stain normal- ization [43]–[46], color augmentation [47], [48], or stain invariant feature learning [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Model-level adaptation is typically done via domain adversarial training [51]–[54], which leverages target domain unlabeled data along with source domain data for modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, the aforementioned data- and model-level adap- tation approaches do not exhaustively assess the gener- alization ability of their trained DL models beyond task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In this case, accurate uncertainty estimation and model calibration are crucial to know when to trust the model – a task known to be challenging for neural networks that often provide over-confident predictions [55]–[57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To the best of our knowledge, computational pathology research in these directions is scarce and remains unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' METHODOLOGY In this section, we present WHOLESIGHT for scalable WSSS of histopathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, we transform a WSI into a TG representation, where nodes and edges of the graph denote tissue regions and their interactions, respec- tively (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Next, a GNN contextualizes node embeddings characterizing tissue regions (Section III-C), which are then processed by a graph classification head for Gleason grading (Section III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Finally, we generate node-level pseudo labels using feature attribution and a node selection strategy, which are used to train a node classifica- tion head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The node-head outputs the segmentation mask with pixel-level Gleason pattern assignment (Section III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' An overview of the method is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Notation and preliminaries We define a graph G ∈ G as (VG, EG, H), where VG and EG denote the set of nodes and edges, respectively, H ∈ R|V |×d denotes d-dimensional node features (or denoted at node-level as Hv,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' := h(v) ∈ Rd), and G is the set of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The neighborhood of a node v ∈ VG is denoted as N(v) := {u ∈ VG | (v, u) ∈ EG ∨ (u, v) ∈ EG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We represent the cardinality of a set as |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='|, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', |N(v)| indicates the number of neighbors of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' GNNs [58] are a class of neural networks that learn from graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, GNNs follow a two- step procedure to contextualize node features by including neighborhood node information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, in an AGGREGATE step, for each node v ∈ VG, the neighboring node features N(v) are aggregated by a differentiable and permutation- invariant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Next, in an UPDATE step, the current features of v and the aggregated features of N(v) are processed by a differentiable operator to update the features of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This procedure is repeated T times, where T is the number of GNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In this work, we use the Graph Isomorphism Network (GIN) [59], where the AGGREGATE step is a mean-operator, and the UPDATE step includes a multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, a GIN layer is given as, h(t+1)(v) = MLP � h(t)(v) + 1 |N(v)| � u∈N (v) h(t)(u) � (1) T GIN layers, denoted as Fθ, are stacked to acquire context information up to T-hops for each v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For graph classi- fication, a fix-sized graph-level embedding hG is derived by pooling the node embeddings hT (v), ∀v ∈ VG by a READOUT step, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', a mean-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Subsequently, hG is mapped to target classes by a classifier network, Fφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similarly, for node classification, hT (v), ∀v ∈ VG can be classified by a classifier network Fψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, classification aims to predict target label y ∈ K for an input x ∈ X, where K and X denote the set of classes and inputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Given a set of sample pairs {(xi, yi)}N i=1, where N is the number of samples and (xi, yi) ∼ p(x, y), the data likelihood can be expressed as p(Y |X, θ, φ) = ΠN i=1p(yi|xi, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The optimal parameters (ˆθ, ˆφ) are obtained by maximum likelihood estimation, or equivalently by minimizing the Negative Log-Likelihood (NLL) − �N i=1 log p(yi|xi, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For graph classification, a sample pair is denoted as (yG, G), yG ∈ KG, G ∈ 3 Figure 1: Overview of the proposed WHOLESIGHT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (a) In the preprocessing step, a TG is constructed to represent a WSI, where the nodes and edges are defined by identifying superpixels and region adjacency connectivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (b) The graph classification head classifies the TG into primary and secondary Gleason patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Subsequently, a feature attribution technique and a node selection strategy derive node-level pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (c) The node classification head learns on the pseudo-labels to classify the nodes, thereby resulting in the WSI segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For node classification, a sample pair is denoted as (yV , v), yV ∈ KV, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For the task in this paper, the set of graph- and node-level classes are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', K := KG = KV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We further introduce the notion of model calibration [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Intuitively, the probability of outcomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', confidence scores, of a calibrated model should match its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For example, the samples predicted with an average confi- dence of 60% by a model should have an average accuracy of 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, for a given network, f : X → K, and p(X, Y ) a joint distribution over the data and the labels, f(x) is said to be calibrated with respect to p if, Ep[Y |f(X) = β] = β, ∀β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The calibration can be visualized with a reliability diagram [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Namely, all the samples in the dataset are assigned to bins according to their predicted confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Then, the model accuracy is computed for the samples in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The network performance is plotted against the binned confidence scores, where deviations from the diagonal represent uncalibrated bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Preprocessing and tissue-graph construction First, we stain-normalize the input H&E stained images using the method by [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It reduces appearance variability across images caused during tissue preparation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', different specimen preparation techniques, staining protocols, fixation characteristics, and imaging device characteristics [62], [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Then, we transform the normalized images into TGs (Fig- ure 1(a)), where the nodes and the edges of a TG denote tissue regions and inter-tissue interactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Mo- tivated by [64], [65], we consider superpixels as the visual primitives to encode tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Compared to rectangular patches, superpixels are more flexible to accommodate ar- bitrary shapes according to the local homogeneity of tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The homogeneity constraint also restricts the superpixels to span across multiple distinct structures and include different morphological regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' TG construction follows [66], where the prominent steps are, (1) detection of superpixels to define nodes VG, (2) characterization of superpixels to define node features H, and (3) building graph topology to define edges EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We 4 WSI Superpixels Tissue-graph P+S: 3+4 classification head Primary Attribution Pseudo-labels REA- Benign (b) Graph G3 G4 GNN MLP G5 D Fe Secondary F Benign Benign U G3 G3 T G4 G4 G5 high G5 ow Segmentation classification head (c) Node GNN MLP Fe Benign G3 G4 G5 Training Frozen Tissue-graph input Graph supervision Node supervisionadopt a two-step process to identify superpixels in a WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, we use Simple Linear Iterative Clustering (SLIC) [67] to produce over-segmented superpixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Over-segmentation is conducted at a low magnification to capture homogeneous regions while offering a good compromise between granu- larity and smoothing-out noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In the second step, the over- segmented superpixels are hierarchically merged according to their channel-wise color similarity at high magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Color similarity is quantified in terms of channel-wise 8-bin color histograms, mean, standard deviation, median, energy, and skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The resulting merged tissue regions form the nodes of the TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The merged superpixels denote morpholog- ically meaningful homogeneous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Additionally, merg- ing reduces the node complexity of the TG, thus enables the scaling of TG to a large WSI and contextualization to distant tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We characterize the TG nodes by morphology and spatial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Considering the potentially arbitrary dimension of superpixels, we use a two-step process to derive morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, we extract patches of size 144×144 pixels from a su- perpixel, resize them to 224×224 size, and encode them into 1280-dimensional features via MobileNetV2 network [68] pre-trained on ImageNet [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Superpixel-level features are computed as the mean of the patch-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Next, we compute spatial features for each node by normalizing the superpixel centroids by the image dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Normaliza- tion ensures the invariability of the spatial features to the varying dimensions of input WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Finally, we define the TG edges by constructing a region adjacency graph topol- ogy [70] using the spatial connectivity of superpixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To this end, we assume that adjacent tissue regions biologically interact the most, and thus should be connected in a TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Contextualization of node embeddings Given a TG, we learn discriminative node embeddings (see Figure 1(b)) by using the node context information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', the tissue microenvironment and the inter-tissue inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, we use GIN [59] denoted as Fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Since GNNs can operate on graphs of arbitrary and varying sizes, they allow to encode histopathology images represented in form of TGs without needing tile-based processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' As the discriminative information of a node relies on its local sub- graph structures and can lie at different abstraction levels in the GNN, we employ a Jumping Knowledge [71] strategy to utilize multi-level node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Namely, the final node-level embedding after T GIN-layers is defined as, h(T )(v) = CONCAT(h(t)(v), ∀t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', T}) (2) where CONCAT denotes a concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WSI classification Following the contextualization of node features, a graph- classification head classifies the TG by using graph-level embeddings hG and graph/image-level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To ob- tain a fix-sized hG, we use a READOUT operation that averages the node embeddings h(T )(v), ∀v ∈ VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Subse- quently, hG is input to a multi-task classifier for primary and secondary Gleason grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, the classifier includes two Multi-Layer Perceptrons (MLPs), denoted as Fφ = {Fφ1, Fφ2}, to individually predict the primary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', the worst Gleason pattern, and secondary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', the second- worst Gleason pattern, in the WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Each MLP solves a multi- class problem with |K| Gleason patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', benign, Grade 3, Grade 4, and Grade 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The final Gleason grade is derived as the sum of the predicted primary and secondary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fθ and Fφ are optimized jointly by minimizing the weighted cross-entropy loss, LG = λLCE(yGP , ˆyGP ) + (1 − λ)LCE(yGS, ˆyGS) (3) where, P and S denote primary and secondary labels of ground truth yG and prediction ˆyG, and λ ∈ [0, 1] is a hyper- parameter balancing the two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Further, during training we introduce class-weights as w := {log( � i Ni Ni ), i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', |K|}}, where Ni is the count of class-wise Gleason patterns in the training WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' These weights take care of the class imbalance in Gleason grading by assigning a higher value to classes with lower frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Weakly supervised semantic segmentation Nodes in a TG identify superpixels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', morphologically homogeneous tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Since each Gleason pattern is characterized by distinct morphological patterns, we assume that each tissue region, depicted by a node, includes a unique Gleason pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Thus, the WSI segmentation task is translated into classifying the nodes of the TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In presence of only image supervision, the node classification is achieved in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, pseudo-node labels are generated by using the image labels, and then, the pseudo labels are used to train a node classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Pseudo node labels: Following WSI classification, a post- hoc feature attribution technique is used to measure the importance of each node towards TG classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specif- ically, we use GRAPHGRAD-CAM [72], [73], an exten- sion of GRAD-CAM [74] to operate with GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Given a graph G, GRAPHGRAD-CAM produces class-wise node attribution maps, Ak, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' These maps highlight the importance ∀v ∈ VG for classifying G into |K|, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Provided the importance scores for v towards |K|, it can be assumed that the label of v is k ∈ K, if the highest importance score corresponds to class k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' At this stage, an argmax operation across Ak, ∀k ∈ K can be considered to classify the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, such node labeling may be suboptimal, because, Some nodes marginally contribute and bear low impor- tance scores ∀k ∈ K for classifying a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, an argmax across the importance scores for a node 5 greedily selects the class with the highest score, even though the node label is not ascertained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' A node highly contributing towards the prediction of a class is not necessarily part of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For example, a node can bear high importance if it provides useful complementary information for tie-breaking or ruling out another class possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, if the set of nodes Vk ⊂ V has high importance scores for class k, the labels of Vk are not ensured to be k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Even, the labels of v ∈ Vk are not guaranteed to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' A class attribution map does not necessarily highlight all the nodes belonging to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Depending on the task complexity, a classifier may utilize only a subset of the informative nodes from a class to predict the graph label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Formally, if the set of nodes Vk ⊂ V have high importance scores for class k, then Vk may not include all the nodes in Vk ⊂ V that have the actual label k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', Vk ⊂ Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In presence of several feature attribution techniques in literature, with different underlying mechanisms, can produce different attribution maps [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Thus, a single attribution technique may not be trusted for score-based node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We, therefore, strategize to use the highlighted nodes by feature attribution as pseudo-labels to train a node-classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For a graph G with Gleason score P+S, P, S ∈ K, we compute node importance scores IP and IS, ∀v ∈ VG using GRAPHGRAD-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' As the scores by GRAPHGRAD-CAM are unbounded, we normalize the scores using min-max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Then, we select the top n% nodes above a threshold t, denoted as VP and VS, where n and t are hyperparameters tuned during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It selects the most informative nodes for downstream node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For a node v ∈ VP and v ∈ VS, we use arg max(IP (v), IS(v)) to ensure VP ∩ VS = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Finally, classes with the highest scores are assigned as pseudo labels y ˜V to the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Pursuing the process for all the TGs in the dataset renders pseudo labels Y ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Node classification: Y ˜V is used to train the node- classification head, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically for a graph G, we get the node embeddings h(T )(v), ∀v ∈ VG using Fˆθ, where ˆθ are the parameters of the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fˆθ is frozen during node classification such that the same GNN backbone is used for both segmentation and classifica- tion, thereby reducing the number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' h(T )(v), ∀v are processed by an MLP Fψ to predict Y ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fψ is trained by optimizing a weighted multi-class cross- entropy objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similar to the graph classification, class- weights are defined as w := {log( � i Ni Ni ), i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', |K|}}, where Ni is the number of annotated nodes of class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The node-wise predicted class labels are used to obtain the final segmentation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Noticeably, WHOLESIGHT does not include any customized post-processing, unlike [26], thus being applicable to various tissues and segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Notably, the graph- and the node-classification heads address complementary tasks for a graph G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', at graph- level and at node-level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Therefore, following the training of Fψ, we unfreeze Fθ, and jointly fine-tune ˆθ and ˆψ with a small learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The complementarity of the tasks provides an additional informative signal to further improve the segmentation and classification performance of WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Datasets We evaluate our method on three datasets containing whole-slide prostate cancer needle biopsies for Gleason pattern segmentation and Gleason grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Gleason patterns include grade 3 (G3)- moderately differentiated nuclei and poorly-formed cribriform glands, grade 4 (G4)- poorly dif- ferentiated nuclei and irregular masses, and grade 5 (G5)- less differentiated nuclei and lack or only occasional glands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Normal glands and non-epithelial tissues are labeled as benign (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Gleason grade depicts the worst (primary, P) and the second-worst (secondary, S) Gleason patterns in a WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dataset details are as follows: Sicap dataset: The dataset [75] contains 18,783 patches of size 512×512 with complete pixel-level annotations and slide-level Gleason grades for 155 WSIs from 95 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The original slides and masks were reconstructed by stitch- ing the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The WSIs were scanned at 40× magnifi- cation by Ventana iS-can Coreo scanner and downsampled to 10× magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The slides were annotated by expert urogenital pathologists at the Hospital Cl´ınico of Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Radboud dataset: [76] includes 5,759 needle biopsies from 1,243 patients at the Radboud University Medical Center, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The slides were scanned with a 3D His- tech Panoramic Flash II 250 scanner at 20× magnification (resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='24µm/pixel) and were downsampled to 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Annotations include WSI Gleason grades and noisy pixel- level Gleason pattern masks, released as part of the Prostate cANcer graDe Assessment (PANDA) challenge [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The masks were cleaned for segmentation using standard image manipulation techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', contextualized noise removal, hole filling, and edge smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In absence of large public datasets with pixel-level annotated prostate cancer WSIs, we used this dataset for developing and evaluating our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Karolinska dataset: The dataset [78] comprises of 5,662 core needle biopsies from 1,222 patients at various hospitals in Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The slides were scanned with a Hamamatsu C9600-12 and an Aperio Scan Scope AT2 scanner at 20× magnification with a pixel resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='45202µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5032µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The biopsies were annotated by an expert uro-pathologist for Gleason grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 6 Figure 2: Gleason grade-wise data distribution across train, validation, and test in Karolinska, Radboud and Sicap datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Each dataset is split into train, validation, and test in a ratio of 60%, 20%, and 20% at Gleason grade level, using a random stratification that preserves the percentage of classes in each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The dataset distributions and splits are displayed in Figure 2, which highlights the class-level imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Implementation and evaluation We implemented WHOLESIGHT using PyTorch [79], DGL [80], and Histocartography [81], and conducted exper- iments on NVIDIA Tesla P100 GPU and POWER9 CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To develop the WHOLESIGHT network, Fθ, Fφ, and Fψ were designed by optimizing their respective hyperpa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, Fθ and Fφ were trained by using image- level labels, and then pseudo-node labels were created to train Fψ to produce segmentation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The number of GIN layers in Fθ were optimized for the values {3, 4, 5}, where the UPDATE function was defined as a 2-layer MLP with 64 hidden units and ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fφ contains two heads for classifying primary and secondary Gleason grades, where each head consists of a 2-layer MLP with 128 hidden units and ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fψ contains a 2-layer MLP with 128 hidden units and ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Considering the small size of the Sicap dataset, node- level augmentations were employed to augment the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, random node rotations {90, 180, 270} degrees, and horizontal and vertical mirroring were used for augmenting the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Batch size and learning rate were optimized from {4, 8, 16} and {10−4, 5×10−4, 10−3}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dropout layers with rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 were included in the MLPs belonging to Fθ, Fφ, and Fψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The pseudo-node labels were extracted for selection percentages in {5, 10, 15, 20} and thresholds {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Following the hyperparameter tuning, ten WHOLESIGHT models were trained with different network initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Validation weighted-F1 was used for model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The reported results correspond to the mean and standard deviation over these ten models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Classification metrics: Classification performance was measured by the weighted-F1 score of Gleason grade and the quadratic kappa score (κ2) of ISUP grade [82], [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' ISUP is an alternate grading system whose correspondence with Gleason grading is defined as, Benign → ISUP-0, GG- (3+3) → ISUP-1, GG-(3+4) → ISUP-2, GG-(4+3) → ISUP- 3, GG-8 → ISUP-4, and GG≥9 → ISUP-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' κ2 captures the degree of disagreement between the prediction and ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For example, a grade 6 sample predicted as grade 10 is penalized more than predicting grade 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Segmentation metrics: Segmentation performance was measured by Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Given the imbalance of the Gleason patterns in the datasets, we also reported the per-pattern Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Uncertainty metrics: Following the work of [84], we evaluated the classification uncertainties in terms of Brier score sB (lower is better) and the NLL sNLL (lower is better) over a set of N test samples, defined as, sB = 1 N N � n=1 |K| � i=1 (yi − ˆyi)2, sNLL = − 1 N N � n=1 |K| � i=1 p(yi) log ˆp(yi) (4) Calibration metrics: Reliability diagrams provide an intu- itive understanding of model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To quantify these observations, we used the Expected Calibration Error (ECE) metric [85], which computes the weighted average deviation of the confidence scores over all the bins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', cECE = B � b=1 Nb N |acc(b) − conf(b)| (5) where nb is the number of samples in bin b, acc(b) and conf(b) are the accuracy and the average confidence of samples in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Baselines We compared WHOLESIGHT with state-of-the-art WSI classification methods and two variants of WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 7 Sicap Radboud Karolinska Train Train Train 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='40 - Val Val Val 1920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='35 - Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='35 Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='35 Test 1812 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='30 - 8 36 36 31 29 963 985 852 860 % % % 764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='15 - 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='10 479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='10 235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='05 109 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='00 - Benign GG6 Benign GG6 GG7 GG7 GG8 GG9 GG10 GG7 GG8 GG9GG10 Benign GG6 GG8 GG9 GG10FSConv: We implemented the two-step method proposed by [75] for WSI classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' First, we extracted patches of size 256×256 from WSIs and classified them using FSConv+global-max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The patches were labeled us- ing the Gleason pattern masks, and patches with >90% homogeneous pattern were selected for classifier training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' During inference, dense patch predictions produced the output segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' An MLP was trained on the Gleason grade percentages over the WSI patches for Gleason grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT(Graph, GRAPHGRAD-CAM): In com- parison to WHOLESIGHT, this baseline contained only Fθ and Fφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It did not create or utilize pseudo labels, and the segmentation output was obtained by taking the argmax over the class-wise GRAPHGRAD-CAM attribution maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT(Multiplex, NC): This variant used both image- and pixel-level supervision during training and acts as the upper bound for WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' As pixel-level annotations were available, Fψ was trained using ground- truth node-level labels, instead of generating pseudo-node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The model consisted of the same Fθ, Fφ, and Fψ as the WHOLESIGHT architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In this baseline, Fθ, Fφ, and Fψ were trained jointly by optimizing a multi- task objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', WSI-level primary and secondary Glea- son score prediction along with node-level Gleason pattern prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This variant of WHOLESIGHT was proposed in our preliminary work, as described in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Multiple Instance Learning (MIL): MIL methods are state-of-the-art for WSI classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In particular, we com- pared to ABMIL [30], which used an attention mechanism to aggregate patch embeddings into a fix-sized WSI embedding that was fed to a classifier for Gleason grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We also in- cluded CLAM [14], a method built on ABMIL by including an additional constrain to cluster similar patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Our experiments followed the public implementations * with adjustments to enable multi-task classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For all the baselines, hyper-parameters are thoroughly tuned to use the best learning rate and batch size, if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Subsequently, ten models were re-trained from scratch with the optimal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We report the mean and standard deviation over these runs for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WSSS performance analysis We studies the classification and segmentation perfor- mance of WHOLESIGHT and the competing methods by independently training and testing them on Sicap, Radboud, and Karolinska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Performance analysis: Table I presents the results on Sicap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The analyses are grouped into two supervision set- tings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', complete (C) and weak (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Setting-C utilizes both image- and pixel-level annotations, whereas, Setting- W only uses image-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT reached CLAM publicly available code: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='com/mahmoodlab/CLAM 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6% average Dice score, which significantly outperforms WHOLESIGHT (Graph, GRAPHGRAD-CAM) by +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6% in absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT (Multiplex, NC), that acts as the upper bound, produced a significant gain in segmentation compared to WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The per-class Dice scores indicate that the benign patterns that constitute most tissue areas have a high detection rate compared to less occurring Gleason patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For the classification task, WHOLESIGHT outperformed ABMIL and CLAM, both in terms of Gleason grade weighted-F1 and ISUP κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Notably, Table II presents the results on Radboud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT rendered an absolute gain of +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8% in average Dice score over WHOLESIGHT (Graph, GRAPHGRAD-CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This confirms the utility of pseudo-node labels for superior segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similar to the observations on Sicap, benign patterns had a high detection rate, followed by G3, G4, and G5 patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' As Radboud dataset includes more G5 patterns than Sicap, we observed a significant gain in detecting high-grade Gleason patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For the classification task, the observations were also consistent with the observations on Sicap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Table III presents the results on Karolinska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In absence of ground truth pixel-level annotations, the segmentation per- formances could not be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT (Graph) outperformed the baselines in terms of classification perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The observations across Table I, III and III conclude that, jointly optimizing classification and segmentation objectives provide complementary information to improve the overall classification performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', WHOLESIGHT (Multiplex) > WHOLESIGHT > WHOLESIGHT (Graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Generalization: performance, uncertainty, and calibra- tion We studied the generalization ability of WHOLESIGHT following a modified training setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, we used Radboud and Karolinska training WSIs for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Thus, the training set encompassed better sample variability and diagnostically more challenging cases than the stan- dalone training counterparts on individual datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Testing was performed individually on Radboud and Karolinska test WSIs, herein studying the in-domain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Further, we tested on the entire Sicap dataset, which consisted of out-of-domain WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Performance analysis: Table IV compared the classifi- cation performance of WHOLESIGHT and the competing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In terms of the weighted-F1 score on the in- domain test set, WHOLESIGHT outperformed ABMIL and performed better or comparable to CLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similar patterns were also observed for ISUP κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, the variances of the classification for WHOLESIGHT is consistently lower than ABMIL and CLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' When tested on the out-of-domain Sicap dataset, WHOLESIGHT achieved significantly better classification than ABMIL and CLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 8 Table I: Classification and segmentation results on Sicap dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The best performances for using image-level supervision are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Annot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' per-class Dice avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dice GG wF1 ISUP κ2 Method Benign Grade3 Grade4 Grade5 C FSConv [75] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 WHOLESIGHT (Multiplex, NC) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 W ABMIL [30] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 CLAM [14] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 WHOLESIGHT 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 (Graph, GRAPHGRAD-CAM) WHOLESIGHT 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 (Graph + Pseudo nodes, Node class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=') Table II: Classification and segmentation results on Radboud dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The best performances for using image-level supervision are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Annot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' per-class Dice avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dice GG wF1 ISUP κ2 Method Benign Grade3 Grade4 Grade5 C FSConv [75] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 WHOLESIGHT (Multiplex, NC) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 W ABMIL [30] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 CLAM [14] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 WHOLESIGHT 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 (Graph, GRAPHGRAD-CAM) WHOLESIGHT 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 (Graph + Pseudo nodes, Node class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=') Table III: Classification results on Karolinska dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The best performances for using image-level supervision are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' GG wF1 ISUP κ2 W ABMIL [30] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 CLAM [14] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 WHOLESIGHT (Graph) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 Table IV: Classification and segmentation results on Radboud, Karolinska, and Sicap datasets for models trained using both Radboud and Karolinska datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Annot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Radboud Karolinska Sicap Method avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dice GG wF1 ISUP κ2 GG wF1 ISUP κ2 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Dice GG wF1 ISUP κ2 C FSConv [75] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 WHOLESIGHT (Multiplex, NC) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8 W ABMIL [30] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8 - 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 CLAM [14] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 - 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 WHOLESIGHT 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 (Graph, GRAD-CAM) WHOLESIGHT 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='1 (Graph + Pseudo nodes, Node class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=') Confusion matrices for the best Gleason grading, ISUP grading, primary- and secondary classification with WHOLESIGHT are presented in Figure 3 on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It can be observed that most misclassifications lie close to the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Majority of the confusion occurred between GG6 and GG7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', GG(3 + 3) versus GG(3 + 4) and GG(4 + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Such ambiguity is prevalent among pathologists, as shown in [86], [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' High-grade Gleason grading better on Radboud than Karolinska due to more number of high-grade samples in Radboud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Primary- and secondary classification weighted-F1 for Radboud, Karolin- ska and Sicap were 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3%, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7%, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6% and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5%, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7%, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This indicated that identifying secondary Gleason pattern is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Table IV 9 Figure 3: Confusion matrices for Gleason grading, ISUP grading, primary- and secondary Gleason classification on Radboud, Karolinska, and Sicap datasets for with the best WHOLESIGHT model trained using Radboud and Karolinska training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' also presents the generalizability assessment of segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT consistently performed better than WHOLESIGHT (Graph, GRAPHGRAD-CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Noticeably, the Dice scores on Radboud and Sicap datasets improved over the segmentation results in Table II and I by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4% for WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It can be reasoned to the usage of more training WSIs, which indicate that WSSS can be improved by utilizing more weak supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Uncertainty analysis: Figure 4 presents the classification uncertainty analysis of WHOLESIGHT, WHOLESIGHT (Graph, GRAD-CAM), and WHOLESIGHT (Multiplex, NC), in terms of NLL and Brier score, on Radboud and Karolinska datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT (Multiplex, NC) ren- dered a significantly lower NLL than WHOLESIGHT across all datasets for primary, secondary, and Gleason grade (P+S) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Noticeably, the NLL and Brier scores were consistently higher for predicting the secondary Gleason patterns than the primary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This resonates with the fact that identifying secondary patterns is more challenging with higher ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Model calibration analysis: A model with good uncer- tainty estimate should be well-calibrated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', the model con- fidence should be close to the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Figure 4 presents the reliability diagrams of the primary classification head on Karolinska and Radboud datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WHOLESIGHT showed consistently better calibration than WHOLESIGHT (Graph, GRAD-CAM) and similar calibration with respect to WHOLESIGHT (Multiplex, NC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' ECE also metric quan- titatively supported this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, we observed that still all models remains over-confident as the model accuracies over the confidence bins remained lower than the expected calibration (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 10 Gleason grade ISUP grade Primary Gleason Secondary Gleason Benign 7 2 5 3 0 Benign 179 7 0 2 5 3 Benign 179 7 10 0 Benign 179 9 5 3 Grade6 89 21 17 3 0 0 Gradel 17 15 6 0 3 Radboud Grade3 - 185 45 label 21 23 197 56 26 label 1 True label Grade7 192 24 22 Grade2 28 52 31 0 10 35 4 3 True Grade8 24 56 24 Grade3 - 7 16 3 93 21 23 4 6 14 20 Grade4- 31 330 Grade4- 82 121 55 9 Grade9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 22 35 75 22 Grade4 - 1 3 5 19 56 32 5 4 32 28 Grade5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Grade5 2 1 21 41 74 5 0 1 22 38 109 Gradel0 - 3 4 8 Grade5 - 6 1 1 ex ade2 ade4 Gra Gr Gr Gr Gr Predicted label Predicted label Predicted label Predicted label 0 Benign 30 2 0 Benign 2 2 343 30 2 343 2 32 0 Benign- 343 Benign 343 30 4 2 4 Gradel 0 Grade6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 253 30 0 0 253 30 60 6 0 6 60 Karolinska 70 378 26 Grade3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 63 274 69 2 True label 1 Grade7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 59 86 21 True label Grade2 - 56 39 13 2 1 10 8 9 rue Grade8 - 27 51 Grade3 - 5 3 21 18 12 6 2 1 3 12 40 Grade4 17 107 10 138 5 89 Grade9 0 8 16 15 Grade4 7 7 20 6 6 2 2 6 19 Grade5- 0 6 Grade5- 2 20 0 Grade10 0 0 0 0 2 Grade5 3 5 16 0 0 24 2 Gr Predicted label Predicted label Predicted label Predicted label Benign Benign 34 1 0 1 0 0 34 0 0 0 1 Benign Benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 34 1 0 34 1 1 0 1 0 Gradel Grade6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 6 0 0 24 1 0 0 5 2 Grade3 - 33 16 Grade3 - 37 0 2 True label Grade7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 19 4 label Grade2 3 5 5 1 0 0 1 0 0 0 Sicap True Grade8 10 15 8 Grade3 1 0 2 7 6 1 12 54 10 14 22 15 5 8 1 Grade4 - 1 ９ 9 Grade9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 0 1 1 1 Grade5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 1 0 7 3 Grade5 2 6 11v 1 0 2 2 2 1 0 1 7 13 Gradel0 - 0 Grade5 - 1 1 e de4 ade5 ade2 e Predicted label Predicted label Predicted label Predicted labelFigure 4: Uncertainty and model calibration analysis of WHOLESIGHT, WHOLESIGHT (Graph, GRAD-CAM), and WHOLESIGHT (Multiplex, NC) models for Radboud (a, b, c) and Karolinska (d, e, f) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (a, d) and (b, e) present NLL (lower is better) and Brier scores (lower is better), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (c, f) present reliability diagrams of the primary Gleason classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Expected calibration (blue) highlights a perfectly calibrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Calibrations of WHOLESIGHT, WHOLESIGHT (Graph, GRAD-CAM), and WHOLESIGHT (Multiplex, NC) are in orange, red, and purple, respectively, along with the number of samples (in %) in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Qualitative analysis We qualitatively analyze the results of WHOLESIGHT by (1) visualizing overlaid segmentation masks on WSIs, (2) analyzing the t-distributed stochastic neighbor (t-SNE) [88] node embeddings, and (3) correlating the segmentation out- puts with pathological reasonings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Segmentation mask visualization: Figure 5 demonstrates segmentation predictions obtained with WHOLESIGHT and WHOLESIGHT(Multiplex, NC) on Sicap dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We can 11 (a) Radboud (d) Karolinska P+$ P s P+$ P s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 - T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 T TTN TIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 (b) (e) P+S P P+S s P s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='6 Brier score Brier score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='2 (c) () 100 100 100 卜100 -○-- Expected calibration --.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Expected calibration -○-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WholeSIGHT Graph- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='27 -○- WholeSIGHT Graph- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='27 -○-- WholeSIGHT- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='23 -○-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WholeSIGHT- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='21 80 - :-○-- WholeSIGHT Multiplex- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='22 80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='-○-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' WholeSIGHT Multiplex- ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='23 80 Accuracy (%) 60 60 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='40 40 / 40 40 20 20 20 20 T0 0 0 50 60 70 80 90 100 50 60 70 80 90 100 Confidence (%) Confidence (%)Figure 5: Sample segmentation maps from Sicap dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Ground truth is shown on the left, WHOLESIGHT predictions in the middle, and WHOLESIGHT(Multiplex, NC) on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Tissue regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', TG nodes, are represented by black overlay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (a, b, c) display GG(3+3), GG(4+4), and GG(5+5) samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For better visualization, benign areas are not highlighted in the segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' 12 (a) Gleason grade 3+3 Benign G3 CA b Gleason grade 4+4 Benign G3 G4 G5 (c) Gleason grade 5+5 Benign G3 G5 Ground truth WholeSIGHT WholeSIGHT (Multiplex, NCFigure 6: t-SNE visualization of tissue-graph node embeddings and example patches from several regions on the two- dimensional t-SNE feature space for Sicap dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (a) t-SNE visualization of the correctly classified nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (b) and (c) display the t-SNE visualization of misclassified nodes, where (b) and (c) highlight the ground truth and predicted node labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (d) and (e) demonstrate square patches of size 224×224 at 10× magnification cropped around the node centroids selected from different regions on the t-SNE embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' (d) and (e) highlight the correctly and incorrectly classified node patches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The labels of the patches in (e) are formatted as Y → ˆY , where Y and ˆY denote the ground truth and the predicted class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The colored rectangles around the patches in (d) and (e) correspond to respective colored rectangles in (a), (b), and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Benign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Correct classifications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Benign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Misclassifications: Ground truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='→G3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='→ G4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Benign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Gleason 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G4→B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='Misclassifications: Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='→ G3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='→ G5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='G5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='100observe that WHOLESIGHT correctly delineates the can- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='cerous regions in the WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Zooming into different regions conclude that the tissue regions of TG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', the nodes of TG, (outlined in black in Figure 5) encode meaningful units of homogeneous tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' It substantiates the relevance of using TG representations for segmenting tissue regions into Gleason patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We further notice that WHOLESIGHT, in a few cases, predicts benign regions adjacent to cancerous patterns as cancerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For example, the benign region, primarily consisting of stroma, in Figure 5(c) is predicted as G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We argue that these false positive detections do not inhibit the applicability of the method, as neighboring cancerous regions are correctly detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' In a few other cases, WHOLESIGHT correctly detects missed cancerous regions in the ground truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For instance, in Figure 5(b), the missing G4 region in the upper part of the WSI is correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Comparing WHOLESIGHT with WHOLESIGHT (Mul- tiplex, NC), we observe that several false positives are re- moved, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', in Figure 5(a), thus offering more accurate seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, the improvements by WHOLESIGHT (Multiplex, NC) are achieved at the cost of training with pixel-level annotations that are hardly available in real- world practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Thus, WHOLESIGHT appears to be an appealing compromise between segmentation performance and annotation requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Visualizing t-SNE feature space: A t-SNE visualization of the learned tissue-level embeddings is demonstrated in Fig- ure 6 for Sicap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' t-SNE projects the GNN node embeddings onto a two-dimensional feature space, allowing to analyze the connection between node embeddings and the Gleason pattern distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Figure 6(a) displays the t-SNE feature space for the cor- rectly classified nodes, which highlights demarcated clusters for each Gleason pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' The large cluster of benign nodes indicates the variability of the benign tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Several patches from each Gleason pattern cluster are presented in Fig- ure 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We can observe the reduced nuclei differentiation across the patches from benign to Gleason grade 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Further, Figure 6(b) and (c) display the t-SNE feature space for the misclassified nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Specifically, Figure 6(b) presents the ground truth node labels, and Figure 6(c) the predicted node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Different embedding locations are further selected and highlighted by different colored rectangles and put in re- lation with corresponding patches to indicate the inter-class ambiguities, as demonstrated in Figure 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For example, the first row in Figure 6(e) showcases patches that are benign but are predicted as G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We can visually compare these patches with the G3 patches in the third row of Figure 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similar ambiguities between other pairs of Gleason patterns are also included in Figure 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Interpreting model outcomes via predicted segmentations: Predicted segmentations provide human-understandable interpretability maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For researchers, the segmentations allow to, (1) identify morphological patterns responsible for WSI classification, (2) analyze failure cases by inspecting pixel-level predictions, and ultimately (3) better understand the model behavior towards biomarker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For pathologists, they assist to, (1) put in relation the predicted WSI-level Gleason scores and the highlighted pixel-level Gleason patterns, (2) confirm that the morphology of the identified cancerous regions aligns with pre-established diagnostic criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Additionally, in the perspective of developing AI-assisted human-in-the-loop tools, a Gleason grading system that can simultaneously classify and segment WSIs is closer to the latest pathological standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Indeed, recent revisions of the Gleason grading system [83] emphasized the importance of reporting the percentage of each grade for better pa- tient stratification and treatment selection [89]–[92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' These percentages can be trivially derived from the predicted segmentation maps by counting the number of pixels be- longing to each pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Naturally, such information is not available in mere WSI classification systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Reporting per- grade percentage is particularly important in ambiguous and borderline cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' For instance, consider two patients with Gleason score 3+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' When a small percentage of pattern- 4 is present, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', 10%, the case can be considered as an intermediate risk cancer where active patient surveillance is enough [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' However, a larger secondary pattern may require specific treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Reporting percentages of each grade allows us to discriminate between these two scenarios easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Similarly, consider a Gleason score 4+3 with a small secondary Gleason pattern, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=', 90% and 10% area for primary and secondary patterns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' This case will be scored as 4+3, even though it is close to a score of 4+4, which would lead to a different treatment protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' By explicitly reporting the Gleason pattern percentages, such corner cases can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' CONCLUSION Accurate delineation of patterns in whole-slide histopathology images typically demands pixel-level annotations, which are hard to acquire in a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Nonetheless, the semantic segmentation of diagnostically relevant patterns is crucial for disease diagnosis and treatment selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To this end, we proposed a novel weakly-supervised semantic segmentation method, WHOLESIGHT, that can segment the relevant patterns of interest in histopathology images by leveraging only image- level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' To our knowledge, WHOLESIGHT is the first weakly-supervised semantic segmentation method that can operate in an end-to-end manner on histopathology images of arbitrary shape and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' We evaluated our proposed method on three publicly available prostate needle biopsy datasets for Gleason grade classification 14 and Gleason pattern segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' On comparing state- of-the-art methods for histopathology applications, we demonstrated the classification and segmentation superiority of WHOLESIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Additionally, WHOLESIGHT is a modular approach that can utilize both image-level and pixel-level supervision to simultaneously perform image classification and segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Though we have evaluated our method for H&E stained prostate cancer needle biopsies, the technology is easily extendable to other tissue types, imaging techniques, and image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Siegel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Miller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Fuchs, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E1T4oBgHgl3EQfIANs/content/2301.02933v1.pdf'} +page_content=' Jemal, “Cancer statis- tics, 2022,” CA: A Cancer Journal for Clinicians, 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0000000000000000000000000000000000000000..0fdace1deef76eac0c895b95e617ff1931d7f495 --- /dev/null +++ b/T9FJT4oBgHgl3EQfMyyd/content/tmp_files/2301.11475v1.pdf.txt @@ -0,0 +1,691 @@ +1 + +The effect of primary school education on preventive behaviours during COVID-19 in +Japan + +Short title: Education and preventive behaviours + +Eiji Yamamura1*, Yoshiro Tsutsui2, Fumio Ohtake3, + + +1 Department of Economics, Seinan Gakuin University, Fukuoka, Japan + +*Corresponding author +Email: yamaei@seinan-gu.ac.jp (EY) +A full list of author information is available at the end of the article + +Abstract +Background +Education plays a critical role on promoting preventive behaviours against the spread +of pandemics. In Japan, hand-washing education in primary schools was positively +correlated with preventive behaviours against COVID-19 transmission for adults in 2020 +during the early stages of COVID-19 [1]. The following year, the Tokyo Olympics were +held in Japan, and a state of emergency was declared several times. Public perceptions of +and risks associated with the pandemic changed drastically with the emergence of +COVID-19 vaccines. We re-examine whether effect of hand-washing education on +preventive behaviours persisted by covering a longer period of the COVID-19 pandemic +than previous studies. + +Methods +26 surveys were conducted nearly once a month for 30 months from March 2020 (the +early stage of COVID-19) to September 2022 in Japan. By corresponding with the same +individuals across surveys, we comprehensively gathered data on preventive behaviours +during this period. In addition, we asked about hand-washing education they had received + +2 + +in their primary school. We used the data to investigate how and the degree to which +school education is associated with pandemic mitigating preventive behaviours. + +Results +We found that hand-washing education in primary school is positively associated with +behaviours such as hand washing and mask wearing as a COVID-19 preventive measure, +but not related to staying at home. We observed a statistically significant difference in +hand washing between adults who received childhood hand-washing education and those +who did not. This difference persisted throughout the study period. In comparison, the +difference in mask wearing between the two groups was smaller, but still statistically +significant. Furthermore, there was no difference in staying at home between them. + +Conclusion +Childhood hygiene education has resulted in individuals engaging in hand washing and +mask wearing to cope with COVID-19. Individuals can form sustainable development- +related habits through childhood education. +Keywords: childhood education, hygiene, COVID-19, preventive behaviours, staying at +home, mask wearing, hand washing, public good + + + + + + + + + + + + + + + + +3 + + + + +Introduction + +Ideally, individuals and organisations would have better preparedness for +unexpected events such as pandemics prior to their emergence. Education, which can +improve preparedness, is expected to alter individuals’ risk perceptions and positively +affect health outcomes. Educational level played critical role in an individual’s +preparedness for the COVID-19 pandemic [2]. Risk perception and precautionary +behaviour against pandemics can be dynamic over time [3]. However, the effect of +education persists for a longer period once hygiene habits are formed [1,4], contributing +to the sustainability of society. +Individuals can cope more smoothly to the occurrence of epidemics and pandemics +if they take appropriate precautions. According to Ikeda et al. [5], individuals in Japan +care about hygiene, such as regular hand washing, reducing the risk of contracting an +infection. Lee et al. explored the role of hygiene education of children in schools on +regular hand-washing behaviours during the COVID-19 pandemic [1]. They found that +hand-washing education in primary school is positively correlated with various +preventive behaviours in adulthood during the COVID-19 pandemic but also prior to the +pandemic. This finding was observed using a dataset collected from April to August 2020 +during the early stage of the COVID-19 pandemic. +According to a study on the 2009 H1N1 influenza pandemic, the Mexican +government promoted campaigns to educate the public about using hand sanitisers, +hand-washing techniques, and wearing masks. Accordingly, past experiences with +pandemics facilitated hand-washing behaviours, which provided long-lasting effects on +health outcomes [4]. The COVID-19 pandemic continued to influence various aspects of +daily life until at least the end of 2022. Individuals learned about COVID-19 based on +their experiences and public campaigns. However, COVID-19 vaccination was +implemented in 2021 throughout Japan, and most individuals have been vaccinated. +Hence, the risk of COVID-19 infection is reduced, which influences the degree of +engagement in preventive behaviours [6–8]. +Overall, the situation changed drastically from the early stages of the COVID-19 +pandemic. It is necessary to consider how and the extent to which the effect of childhood +education on preventive behaviours changed. In this short note, we used monthly + +4 + +individual-level longitudinal data to re-examine the findings of Lee et al. over a longer +time period [1]. We asked: How did hygiene practice education in childhood influence +preventive behaviours in adulthood during the COVID-19 pandemic from March 2020 to +September 2022? +Materials and methods +Data collection +Shortly after COVID-19 infection was detected in Japan in January 2020, we +decided to collect data through online surveys by commissioning the research company +INTAGE. INTAGE was chosen for their good reputation due to their abundant experience +with academic research. In the early stages of the COVID-19 pandemic (March 13–16, +2020), the first wave of queries was conducted to gather 4,359 observations. INTAGE +recruited participants for the survey from among pre-registered individuals, with a +participation rate of 54.7 %. Respondents were randomly selected to fill the pre-specified +quotas by identifying a representative of the Japanese adult population (ages 18–78 years), +and data was collected on household income, age, gender, educational background, and +area of residence. This sampling method was chosen because individuals aged 17 years +and below were too young to be registered with INTAGE, and data from individuals over +78 years of age could not be reliably collected mainly because they were unlikely to use +the Internet. Consequently, the sample population was restricted to ages 18–78 years. +Longitudinal panel data were constructed as follows. Internet surveys were conducted +nearly monthly on 26 occasions (‘waves’) between March 2020 and September 2022 with +the same individuals. Surveys were not conducted for three months between July- +September 2020 because of a shortage of research funds. After acquiring additional +funding, surveys continued in October 2020 (6th wave) and included an additional +question on primary school education to examine the effect of childhood education on +preventive measures. The first survey by Lee et al. was conducted between April 28–30 +[1]. By comparison, we conducted our first survey one month earlier, between March 13– +27. From March to April 2020, the COVID-19 situation in Japan changed drastically, +making this a notable distinction [9–12]. During the study period, some respondents +stopped taking the surveys and were removed from the sample pool. We limited samples +used for analysis to respondents who participated from the first to the 26th wave to follow +the same individuals. Further, we restricted the sample to those who answered various +questions, such as primary household income, job type, and education in primary school. +In particular, many respondents did not remember experiencing hand-washing education + +5 + +in primary school. Eventually, the number of respondents was reduced to 996, and the +total number of observations used in this study was 25,896. + +Methods +Table 1 describes the key variables used in the estimation and reports their means and +standard deviations. The survey questionnaire contained basic questions about +demographics, such as birth year, gender, educational background, household income, +and jobs. + +Table 1. Definitions of key variables. +Variables +Definition + + + + Outcome variables +Mean +s.d. +STAYING +HOME +In the last week, how consistent were you at ‘not going out +of home’? Please choose among 5 choices. +1 (not completed at all) to 5 (completely consistent). +4.21 +0.91 +WEARING +MASK + + +In the last week, how consistent were you at ‘wearing a +mask’? Please choose among 5 choices. +1 (not completed at all) to 5 (completely achieved). +4.54 +0.19 +HAND +WASHING + +In the last week, how consistent were you at ‘washing your +hands’? Please choose among 5 choices. +1 (not completed at all) to 5 (completely achieved). +2.91 +1.29 + + Confounders (Independent variables) + + +WASHING +EDUCATION + +Did everyone in your class was supervised by teachers to +ensure that they washed their hands in turn? +1 (Yes) or 0 (No) +0.48 +0.49 +SCHOOL +UNIFORM +Did you wear school uniforms in primary school? +1 (Yes) or 0 (No) +0.20 +0.40 + +The estimated function takes the following form: +Yit = α0 + α1 WASHING EDUCATIONit + α2 SCHOOL UNIFORMit + kt + uit +Yit is the outcome variable for individual i and wave t and α denotes the regression +parameters. uit is the error term. The estimation method was the ordinary least squares +model. The behaviour of individuals depends on the situation. For instance, residents were +strongly requested to stay at home during states of emergency. There were also cycles of +increasing and decreasing numbers of new infections, which were common in all parts of +Japan [11,12]. kt represents the characteristics of the situation at each time point. To +control for this, we used 25 time point dummies. +Y is the outcome variable captured by the three proxy variables STAYING HOME, +HAND WASHING, and WEARING MASK. The respondents were asked the following +questions about preventive behaviours: + +6 + +‘Within a week, to what degree have you practiced the following behaviours? +Please answer based on a scale of 1 (I have not practiced this behaviour at all) to 5 (I have +completely practiced this behaviour)’. +(1) Staying home +(2) Wearing a mask +(3) Washing my hands thoroughly + +The answers to these questions served as proxies for the following variables for +preventive behaviours: staying home, frequency of hand washing, and degree of mask +wearing. Larger values indicate that respondents are more likely to engage in preventive +behaviours. +The key confounding variable is WASHING EDUCATION, which is ‘1’ if teachers +supervised pupils to ensure that they washed their hands during primary school, otherwise +it is ‘0’. In this study 48% of respondents had experienced hand-washing practice in +primary school (Table 1). Previous studies have found that the experience of wearing +school uniforms during primary school is positively correlated with pro-social +inclinations in adulthood [13]. Therefore, the experience of school uniforms may be +correlated with preventive behaviours in adulthood, so SCHOOL UNIFORM is also +included as a confounding variable. The statistical software used in this study was +Stata/MP 15.0. + +Results +Baseline estimations +Table 2. Dependent variables are preventive behaviours (Data: 1st to 26th wave). + + (1) +HAND +WASHING + (2) +WEARING +MASK + (3) +STAYING +HOME +WASHING EDUCATION +0.197*** +(0.126-0.267) +0.109*** +(0.051-0.167) +0.072 +(−0.032-0.186) +SCHOOL UNIFORM + +0.026 +(−0.067- 0.119) +0.026 +(−0.065-0.061) +−0.002 +(−0.193-0.100) +Time Fixed Effects + Yes +Yes + Yes +Control variables + + Yes +Yes + Yes +Adj R2 +Obs. +0.11 +25,896 +0.19 +25,896 +0.14 +25,896 +Note: Numbers without parentheses are coefficients of the confounding variables. Numbers within +parentheses are 95% CI. The model includes various control variables, such as age, gender, dummies + +7 + +for household income, job dummies, number of deaths, and number of infected people in residential +prefectures. However, these results have not been reported. “Yes” means that these variables are +included. +***p<0.01 + +Table 2 shows the estimation results and coefficient of confounders. We found a positive +correlation for WASHING EDUCATION for all dependent variables. The relationship +between WASHING EDUCATION and both HAND WASHING and WEARING MASK +were found to be statistically significant, whereas STAYING HOME is not. The +coefficient of HAND WASHING is 0.198, meaning that those who experienced hand- +washing education in primary school are more likely to wash their hands by 1.987 points +on a 5-point scale compared to those who did not. The effect of hand-washing education +on HAND WASHING was approximately two times larger than that of WEARING +MASK (0.109). We did not find statistical significance for SCHOOL UNIFORM on any +dependent variable. + Overall, hand-washing education in childhood promotes the hygiene practice of hand +washing and wearing masks, but did not promote staying at home. Table 3 shows that the +results of waves 1–5 are almost identical to those in Table 2. +Table 3. Dependent variables are preventive behaviours (Data: 1st to 5th wave). + + (1) +HAND +WASHING + (2) +WEARING +MASK + (3) +STAYING +HOME +WASHING EDUCATION +0.196*** +(0.103-0.289) +0.121* +(−0.008-0.251) +0.037 +(−0.069-0.136) +SCHOOL UNIFORM + +−0.029 +(−0.127- 0.069) +−0.069 +(−0.190-0.051) +−0.013 +(−0.135-0.109) +Time Fixed Effects + Yes +Yes + Yes +Control variables + + Yes +Yes + Yes +Adj R2 +Obs. +0.11 +4,980 +0.17 +4,980 +0.14 +4,980 +Note: Numbers without parentheses are coefficients of the confounding variables. Numbers within +parentheses are 95% CI. The model includes various control variables, such as age, gender, dummies +for household income, job dummies, number of deaths, and infected persons in residential prefectures. +However, these results have not been reported. “Yes” means that these variables are included. +*p<0.10 +***p<0.01 +Changes of preventive behaviours +Figs 1–3 show that preventive behaviours drastically increased during the early stage + +8 + +of COVID-19, especially during the first state of emergency, as indicated by the solid +vertical line. Subsequently, in Figs 1 and 2, hand washing and mask wearing were +maintained at high levels throughout the study period. This is in contrast with the findings +in Japan that precautionary behaviour in response to the 2009 (H1N1) influenza +pandemic fluctuated [3]. +Those who experienced hand-washing practices in primary school were more likely +to engage in hand washing and mask wearing during the COVID-19 pandemic. The effect +of hand-washing education on mask wearing was smaller and less statistically significant +than that on hand-washing. This may be because hand-washing education is more likely +to form a habit of washing hands than wearing masks. Wearing masks in crowded places +is effective in mitigating pandemics, whereas wearing masks in open air is much less +effective [14]. People wear masks outdoors, partly because of peer pressure. +Hand-washing education played a critical role in forming lasting habits of health- +protective behaviours such as hands-washing and mask wearing. By contrast, Figure 3 +shows the fluctuating cycles of staying at home. Furthermore people became overall less +likely to stay home after the COVID-19 vaccine was implemented, as indicated by the +dashed vertical line. People are unlikely to form a habit of staying home, which is +congruent with Ibuka et al. [3]. There was no difference in staying home between those +who had experienced hygiene education practice and those who did not. + +Fig 1. Hand-washing behaviour +Note: The solid vertical line indicates when the first state of emergency was declared in +Japan. The dashed vertical line shows when COVID-19 vaccination was implemented. + +3.6 +3.8 +4 +4.2 +4.4 +0 +5 +10 +15 +20 +25 +Waves (Mar 2020- Sep 2022) +Washing education +No washing education +Error bar (CI 5%) + +9 + + +Fig 2. Mask-wearing behaviour +Note: The solid vertical line indicates when the first state of emergency was declared in +Japan. The dashed vertical line shows when COVID-19 vaccination was implemented. + +Fig 3. Staying at home behaviour +Note: The solid vertical line indicates when the first state of emergency was declared in +Japan. The dashed vertical line shows when COVID-19 vaccination was implemented. + +3 +3.5 +4 +4.5 +5 +0 +5 +10 +15 +20 +25 +Waves (Mar 2020 - Sep 2022) +Wasing education +No washing education +Error bar (CI 5%) +2.5 +3 +3.5 +0 +5 +10 +15 +20 +25 +Waves (Mar 2020 - Sep 2022) +Washing education +No washing education +Error bars (CI 5%) + +10 + +Discussion +The purpose of this study is to consider how school practices in primary schools +influenced preventive behaviours during the COVID-19 pandemic using data covering +March 2020 to September 2022. Preventive behaviours reduce one’s own risk of being +infected, but also the risk of infecting others. Therefore, preventive behaviours against +pandemic spread can also be considered an investment in public goods to benefit society +[15]. Lee et al. found that hand washing led people to display preventive behaviours even +before COVID-19 (Appendix 7) [1]. Considering their and our findings together, hygiene +education resulted in a habit of hygiene preventive behaviours and persisted regardless +of pandemic severity. +Lee et al. [1] found that hand-washing education is positively associated with various +preventive behaviours, including wearing masks and staying at home. In contrast to Lee +et al., this study found clear differences in educational impact according to the type of +preventive behaviour. In the questionnaire used by Lee et al., detailed questions about +primary school education and various preventive behaviours were included. The +respondents may have perceived the researchers’ intentions, which may have influenced +their responses. For example, their questions about hygiene practice in primary school +may have functioned as a ‘nudge’ that unintentionally influenced respondents to meet the +goals of the researchers and respond accordingly. In our study, questions about preventive +behaviours were included in all waves, whereas questions about primary school education +were blended into various questions only in the 6th wave onward. Hence, before the 6th +wave, respondents may not have perceived our goals to associate childhood education +with preventive behaviours. In order to directly compare our results with those of Lee et +al., we analysed data from the 1st through 5th waves which are almost equivalent to the +period they studied, conducting estimations using the same specifications (Table 3). + Staying at home was not significantly correlated with hand-washing education +during childhood. This might be because staying at home is a different type of preventive +behaviour than hand washing. People stay home only when their benefits exceed their +costs. People sacrifice various experiences through outdoor activities in the real world if +they stay at home. In economic terms, this sacrifice is considered an ‘opportunity cost’ of +staying home. As the opportunity cost is not reduced even if one experiences hand- +washing education in childhood, individuals will stay at home only if their benefits +outweigh their costs regardless of hygiene education. Additionally, staying at home +weakens social ties and reduces social capital because of a reduction in social interaction +through face-to-face communication. As is widely acknowledged, social ties and social + +11 + +capital are positively associated with health status [16–18]. Therefore, it is important to +distinguish staying at home from other preventive behaviours. +The formation of hand-washing habits through hygiene education in childhood reduces +its psychological costs. In this case, people do not need to change their lifestyle to engage +in basic preventive behaviours such as hand washing, regardless of the severity of the +pandemic. Basic hygiene practices in childhood have reduced stress in life during the +pandemic. + +Strength +We constructed longitudinal data to cover a longer period than previous studies in Japan, +where preventive behaviours were not enforced with penalties[1,3]. Lee et al. did not +examine the effects of the emergence and spread of the COVID-19 vaccine on preventive +behaviours [1]. Preventive behaviours of individuals were thought to change in response +to the emergence of the COVID-19 vaccine. However, individuals continue to wash their +hands and wear masks long after vaccine implementation. This clearly suggests these +preventive behaviours are stable. + +Limitation + Many respondents did not remember experiencing hand-washing education in +primary school. We have deleted them from the data pool used for analysis. There was a +difference in the characteristics of respondents who answered the questionnaire and those +who did not. This may have resulted in selection biases. Furthermore, answers to the +questionnaire seem to depend not only on the facts, but also on the respondent's +misapprehension. Therefore, recall bias may occur. Another variable of school education +would show statistical significance if biases had a significant effect on the results. +However, SCHOOL UNIFORM is not significantly correlated with preventive +behaviours, which is clearly different from the results of WASHING EDUCATION. This +suggests, to a certain extent, the biases are minor. +Wearing masks are less effective in open air than indoors [14]. In contrast to Lee et +al. [1], we used only three proxies for preventive behaviours. Therefore, we did not +scrutinise how hand washing and mask wearing changed in different situations. +In contrast to hand washing, the benefit of mask wearing depends on the situation. +Wearing masks in the open air has limited effectiveness [14]. In mid-summer, wearing +masks increased the risk of heatstroke. In this situation, the cost of wearing a mask is + +12 + +higher than its benefits. It is, therefore, important to examine mask-wearing behaviour +in various situations in future studies. + +Conclusion +Preventive behaviours play a vital role in coping with unexpected pandemics such as +COVID-19. We concluded that people can form sustainable development-related habits +through childhood hygiene practice education. + +Acknowledgements +We would like to thank Editage (http://www.editage.com) for their English +Language editing and reviewing of this manuscript. + +Authors’ contributions +EY and FO participated in the conceptualisation of the study and analysed the patient data. +YT designed the panel survey and performed data collection. EY wrote the main text and +made the tables for the original manuscript. All authors reviewed, edited, and approved +the final manuscript. The authors are responsible for any errors in this study. + +Funding +This study was supported by the Fostering Joint International Research B (Grant No. +18KK0048) and the Grant-in-Aid for Scientific Research S (Grant No. 20H05632) from +the Japan Society for the Promotion of Science to Yoshiro Tsutsui and Fumio Ohtake, +respectively. + + +Availability of data and materials +The datasets used and analysed in this study are available from the corresponding author +upon reasonable request. + +Declarations + +13 + +Ethics approval and consent to participate +This study was conducted with the ex-ante approval of the Ethics Committee of the +Graduate School of Economics, Osaka University, and all methods were carried out in +accordance with the relevant guidelines and regulations. The ethics approval number of +Osaka University for this study is R021014. Informed consent for study participation was +obtained from all subjects. + +Consent for publication +Not applicable. + +Competing interests +The authors declare that they have no competing interests. + +Author details +1. Department of Economics, Seinan Gakuin University, Fukuoka, Japan. 2. Faculty of +Social Relations, Kyoto Bunkyo University, Kyoto, JAPAN, 3. Center for Infectious +Disease Education and Research (CiDER), Osaka University, Osaka, Japan. + +References + +1. +Lee, S.Y.; Sasaki, S.; Kurokawa, H.; Ohtake, F. The School Education, Ritual +Customs, and Reciprocity Associated with Self-Regulating Hand Hygiene +Practices during COVID-19 in Japan. BMC Public Health 2022, 22, +doi:10.1186/s12889-022-14012-z. +2. +Cvetković, V.M.; Nikolić, N.; Radovanović Nenadić, U.; Öcal, A.; K. Noji, E.; +Zečević, M. Preparedness and Preventive Behaviors for a Pandemic Disaster +Caused by COVID-19 in Serbia. 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J Urban Health 2001, 78, 458–467, +doi:10.1093/jurban/78.3.458. + + diff --git a/T9FJT4oBgHgl3EQfMyyd/content/tmp_files/load_file.txt b/T9FJT4oBgHgl3EQfMyyd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c25641593435712d24025e4e1df675c061283bd7 --- /dev/null +++ b/T9FJT4oBgHgl3EQfMyyd/content/tmp_files/load_file.txt @@ -0,0 +1,536 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf,len=535 +page_content='1 The effect of primary school education on preventive behaviours during COVID-19 in Japan Short title: Education and preventive behaviours Eiji Yamamura1*, Yoshiro Tsutsui2, Fumio Ohtake3, 1 Department of Economics, Seinan Gakuin University, Fukuoka, Japan Corresponding author Email: yamaei@seinan gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='jp (EY) A full list of author information is available at the end of the article Abstract Background Education plays a critical role on promoting preventive behaviours against the spread of pandemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In Japan, hand-washing education in primary schools was positively correlated with preventive behaviours against COVID-19 transmission for adults in 2020 during the early stages of COVID-19 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The following year, the Tokyo Olympics were held in Japan, and a state of emergency was declared several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Public perceptions of and risks associated with the pandemic changed drastically with the emergence of COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We re-examine whether effect of hand-washing education on preventive behaviours persisted by covering a longer period of the COVID-19 pandemic than previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Methods 26 surveys were conducted nearly once a month for 30 months from March 2020 (the early stage of COVID-19) to September 2022 in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' By corresponding with the same individuals across surveys, we comprehensively gathered data on preventive behaviours during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In addition, we asked about hand-washing education they had received 2 in their primary school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We used the data to investigate how and the degree to which school education is associated with pandemic mitigating preventive behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Results We found that hand-washing education in primary school is positively associated with behaviours such as hand washing and mask wearing as a COVID-19 preventive measure, but not related to staying at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We observed a statistically significant difference in hand washing between adults who received childhood hand-washing education and those who did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This difference persisted throughout the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In comparison, the difference in mask wearing between the two groups was smaller, but still statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Furthermore, there was no difference in staying at home between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Conclusion Childhood hygiene education has resulted in individuals engaging in hand washing and mask wearing to cope with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Individuals can form sustainable development- related habits through childhood education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Keywords: childhood education, hygiene, COVID-19, preventive behaviours, staying at home, mask wearing, hand washing, public good 3 Introduction Ideally, individuals and organisations would have better preparedness for unexpected events such as pandemics prior to their emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Education, which can improve preparedness, is expected to alter individuals’ risk perceptions and positively affect health outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Educational level played critical role in an individual’s preparedness for the COVID-19 pandemic [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Risk perception and precautionary behaviour against pandemics can be dynamic over time [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, the effect of education persists for a longer period once hygiene habits are formed [1,4], contributing to the sustainability of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Individuals can cope more smoothly to the occurrence of epidemics and pandemics if they take appropriate precautions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' According to Ikeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' [5], individuals in Japan care about hygiene, such as regular hand washing, reducing the risk of contracting an infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' explored the role of hygiene education of children in schools on regular hand-washing behaviours during the COVID-19 pandemic [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' They found that hand-washing education in primary school is positively correlated with various preventive behaviours in adulthood during the COVID-19 pandemic but also prior to the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This finding was observed using a dataset collected from April to August 2020 during the early stage of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' According to a study on the 2009 H1N1 influenza pandemic, the Mexican government promoted campaigns to educate the public about using hand sanitisers, hand-washing techniques, and wearing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Accordingly, past experiences with pandemics facilitated hand-washing behaviours, which provided long-lasting effects on health outcomes [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The COVID-19 pandemic continued to influence various aspects of daily life until at least the end of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Individuals learned about COVID-19 based on their experiences and public campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, COVID-19 vaccination was implemented in 2021 throughout Japan, and most individuals have been vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Hence, the risk of COVID-19 infection is reduced, which influences the degree of engagement in preventive behaviours [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Overall, the situation changed drastically from the early stages of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' It is necessary to consider how and the extent to which the effect of childhood education on preventive behaviours changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In this short note, we used monthly 4 individual-level longitudinal data to re-examine the findings of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' over a longer time period [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We asked: How did hygiene practice education in childhood influence preventive behaviours in adulthood during the COVID-19 pandemic from March 2020 to September 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Materials and methods Data collection Shortly after COVID-19 infection was detected in Japan in January 2020, we decided to collect data through online surveys by commissioning the research company INTAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' INTAGE was chosen for their good reputation due to their abundant experience with academic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In the early stages of the COVID-19 pandemic (March 13–16, 2020), the first wave of queries was conducted to gather 4,359 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' INTAGE recruited participants for the survey from among pre-registered individuals, with a participation rate of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='7 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Respondents were randomly selected to fill the pre-specified quotas by identifying a representative of the Japanese adult population (ages 18–78 years), and data was collected on household income, age, gender, educational background, and area of residence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This sampling method was chosen because individuals aged 17 years and below were too young to be registered with INTAGE, and data from individuals over 78 years of age could not be reliably collected mainly because they were unlikely to use the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Consequently, the sample population was restricted to ages 18–78 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Longitudinal panel data were constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Internet surveys were conducted nearly monthly on 26 occasions (‘waves’) between March 2020 and September 2022 with the same individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Surveys were not conducted for three months between July- September 2020 because of a shortage of research funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' After acquiring additional funding, surveys continued in October 2020 (6th wave) and included an additional question on primary school education to examine the effect of childhood education on preventive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The first survey by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' was conducted between April 28–30 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' By comparison, we conducted our first survey one month earlier, between March 13– 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' From March to April 2020, the COVID-19 situation in Japan changed drastically, making this a notable distinction [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' During the study period, some respondents stopped taking the surveys and were removed from the sample pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We limited samples used for analysis to respondents who participated from the first to the 26th wave to follow the same individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Further, we restricted the sample to those who answered various questions, such as primary household income, job type, and education in primary school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In particular, many respondents did not remember experiencing hand-washing education 5 in primary school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Eventually, the number of respondents was reduced to 996, and the total number of observations used in this study was 25,896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Methods Table 1 describes the key variables used in the estimation and reports their means and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The survey questionnaire contained basic questions about demographics, such as birth year, gender, educational background, household income, and jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Definitions of key variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Variables Definition Outcome variables Mean s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' STAYING HOME In the last week, how consistent were you at ‘not going out of home’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 1 (not completed at all) to 5 (completely consistent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='91 WEARING MASK In the last week, how consistent were you at ‘wearing a mask’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 1 (not completed at all) to 5 (completely achieved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='19 HAND WASHING In the last week, how consistent were you at ‘washing your hands’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 1 (not completed at all) to 5 (completely achieved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='29 Confounders (Independent variables) WASHING EDUCATION Did everyone in your class was supervised by teachers to ensure that they washed their hands in turn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 1 (Yes) or 0 (No) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='49 SCHOOL UNIFORM Did you wear school uniforms in primary school?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 1 (Yes) or 0 (No) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='40 The estimated function takes the following form: Yit = α0 + α1 WASHING EDUCATIONit + α2 SCHOOL UNIFORMit + kt + uit Yit is the outcome variable for individual i and wave t and α denotes the regression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' uit is the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The estimation method was the ordinary least squares model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The behaviour of individuals depends on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' For instance, residents were strongly requested to stay at home during states of emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' There were also cycles of increasing and decreasing numbers of new infections, which were common in all parts of Japan [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' kt represents the characteristics of the situation at each time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' To control for this, we used 25 time point dummies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Y is the outcome variable captured by the three proxy variables STAYING HOME, HAND WASHING, and WEARING MASK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The respondents were asked the following questions about preventive behaviours: 6 ‘Within a week, to what degree have you practiced the following behaviours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Please answer based on a scale of 1 (I have not practiced this behaviour at all) to 5 (I have completely practiced this behaviour)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' (1) Staying home (2) Wearing a mask (3) Washing my hands thoroughly The answers to these questions served as proxies for the following variables for preventive behaviours: staying home, frequency of hand washing, and degree of mask wearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Larger values indicate that respondents are more likely to engage in preventive behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The key confounding variable is WASHING EDUCATION, which is ‘1’ if teachers supervised pupils to ensure that they washed their hands during primary school, otherwise it is ‘0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In this study 48% of respondents had experienced hand-washing practice in primary school (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Previous studies have found that the experience of wearing school uniforms during primary school is positively correlated with pro-social inclinations in adulthood [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Therefore, the experience of school uniforms may be correlated with preventive behaviours in adulthood, so SCHOOL UNIFORM is also included as a confounding variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The statistical software used in this study was Stata/MP 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Results Baseline estimations Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Dependent variables are preventive behaviours (Data: 1st to 26th wave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' (1) HAND WASHING (2) WEARING MASK (3) STAYING HOME WASHING EDUCATION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='197** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='267) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='109** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='167) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='072 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='186) SCHOOL UNIFORM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='026 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='119) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='026 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='061) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='002 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='100) Time Fixed Effects Yes Yes Yes Control variables Yes Yes Yes Adj R2 Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='11 25,896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='19 25,896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='14 25,896 Note: Numbers without parentheses are coefficients of the confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Numbers within parentheses are 95% CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The model includes various control variables, such as age, gender, dummies 7 for household income, job dummies, number of deaths, and number of infected people in residential prefectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, these results have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' “Yes” means that these variables are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='01 Table 2 shows the estimation results and coefficient of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We found a positive correlation for WASHING EDUCATION for all dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The relationship between WASHING EDUCATION and both HAND WASHING and WEARING MASK were found to be statistically significant, whereas STAYING HOME is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The coefficient of HAND WASHING is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='198, meaning that those who experienced hand- washing education in primary school are more likely to wash their hands by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='987 points on a 5-point scale compared to those who did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The effect of hand-washing education on HAND WASHING was approximately two times larger than that of WEARING MASK (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We did not find statistical significance for SCHOOL UNIFORM on any dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Overall, hand-washing education in childhood promotes the hygiene practice of hand washing and wearing masks, but did not promote staying at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Table 3 shows that the results of waves 1–5 are almost identical to those in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Dependent variables are preventive behaviours (Data: 1st to 5th wave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' (1) HAND WASHING (2) WEARING MASK (3) STAYING HOME WASHING EDUCATION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='196** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='289) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='121 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='251) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='037 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='136) SCHOOL UNIFORM −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='029 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='069) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='069 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='051) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='013 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='109) Time Fixed Effects Yes Yes Yes Control variables Yes Yes Yes Adj R2 Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='11 4,980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='17 4,980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='14 4,980 Note: Numbers without parentheses are coefficients of the confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Numbers within parentheses are 95% CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The model includes various control variables, such as age, gender, dummies for household income, job dummies, number of deaths, and infected persons in residential prefectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, these results have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' “Yes” means that these variables are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='10 ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='01 Changes of preventive behaviours Figs 1–3 show that preventive behaviours drastically increased during the early stage 8 of COVID-19, especially during the first state of emergency, as indicated by the solid vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Subsequently, in Figs 1 and 2, hand washing and mask wearing were maintained at high levels throughout the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This is in contrast with the findings in Japan that precautionary behaviour in response to the 2009 (H1N1) influenza pandemic fluctuated [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Those who experienced hand-washing practices in primary school were more likely to engage in hand washing and mask wearing during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The effect of hand-washing education on mask wearing was smaller and less statistically significant than that on hand-washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This may be because hand-washing education is more likely to form a habit of washing hands than wearing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Wearing masks in crowded places is effective in mitigating pandemics, whereas wearing masks in open air is much less effective [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' People wear masks outdoors, partly because of peer pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Hand-washing education played a critical role in forming lasting habits of health- protective behaviours such as hands-washing and mask wearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' By contrast, Figure 3 shows the fluctuating cycles of staying at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Furthermore people became overall less likely to stay home after the COVID-19 vaccine was implemented, as indicated by the dashed vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' People are unlikely to form a habit of staying home, which is congruent with Ibuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' There was no difference in staying home between those who had experienced hygiene education practice and those who did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Hand-washing behaviour Note: The solid vertical line indicates when the first state of emergency was declared in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The dashed vertical line shows when COVID-19 vaccination was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='8 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='4 0 5 10 15 20 25 Waves (Mar 2020 Sep 2022) Washing education No washing education Error bar (CI 5%) 9 Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Mask-wearing behaviour Note: The solid vertical line indicates when the first state of emergency was declared in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The dashed vertical line shows when COVID-19 vaccination was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Staying at home behaviour Note: The solid vertical line indicates when the first state of emergency was declared in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The dashed vertical line shows when COVID-19 vaccination was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='5 5 0 5 10 15 20 25 Waves (Mar 2020 Sep 2022) Wasing education No washing education Error bar (CI 5%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='5 0 5 10 15 20 25 Waves (Mar 2020 Sep 2022) Washing education No washing education Error bars (CI 5%) 10 Discussion The purpose of this study is to consider how school practices in primary schools influenced preventive behaviours during the COVID-19 pandemic using data covering March 2020 to September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Preventive behaviours reduce one’s own risk of being infected, but also the risk of infecting others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Therefore, preventive behaviours against pandemic spread can also be considered an investment in public goods to benefit society [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' found that hand washing led people to display preventive behaviours even before COVID-19 (Appendix 7) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Considering their and our findings together, hygiene education resulted in a habit of hygiene preventive behaviours and persisted regardless of pandemic severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' [1] found that hand-washing education is positively associated with various preventive behaviours, including wearing masks and staying at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In contrast to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=', this study found clear differences in educational impact according to the type of preventive behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In the questionnaire used by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=', detailed questions about primary school education and various preventive behaviours were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The respondents may have perceived the researchers’ intentions, which may have influenced their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' For example, their questions about hygiene practice in primary school may have functioned as a ‘nudge’ that unintentionally influenced respondents to meet the goals of the researchers and respond accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In our study, questions about preventive behaviours were included in all waves, whereas questions about primary school education were blended into various questions only in the 6th wave onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Hence, before the 6th wave, respondents may not have perceived our goals to associate childhood education with preventive behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In order to directly compare our results with those of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=', we analysed data from the 1st through 5th waves which are almost equivalent to the period they studied, conducting estimations using the same specifications (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Staying at home was not significantly correlated with hand-washing education during childhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This might be because staying at home is a different type of preventive behaviour than hand washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' People stay home only when their benefits exceed their costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' People sacrifice various experiences through outdoor activities in the real world if they stay at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In economic terms, this sacrifice is considered an ‘opportunity cost’ of staying home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' As the opportunity cost is not reduced even if one experiences hand- washing education in childhood, individuals will stay at home only if their benefits outweigh their costs regardless of hygiene education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Additionally, staying at home weakens social ties and reduces social capital because of a reduction in social interaction through face-to-face communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' As is widely acknowledged, social ties and social 11 capital are positively associated with health status [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Therefore, it is important to distinguish staying at home from other preventive behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The formation of hand-washing habits through hygiene education in childhood reduces its psychological costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In this case, people do not need to change their lifestyle to engage in basic preventive behaviours such as hand washing, regardless of the severity of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Basic hygiene practices in childhood have reduced stress in life during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Strength We constructed longitudinal data to cover a longer period than previous studies in Japan, where preventive behaviours were not enforced with penalties[1,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' did not examine the effects of the emergence and spread of the COVID-19 vaccine on preventive behaviours [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Preventive behaviours of individuals were thought to change in response to the emergence of the COVID-19 vaccine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, individuals continue to wash their hands and wear masks long after vaccine implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This clearly suggests these preventive behaviours are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Limitation Many respondents did not remember experiencing hand-washing education in primary school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We have deleted them from the data pool used for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' There was a difference in the characteristics of respondents who answered the questionnaire and those who did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This may have resulted in selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=" Furthermore, answers to the questionnaire seem to depend not only on the facts, but also on the respondent's misapprehension." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Therefore, recall bias may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Another variable of school education would show statistical significance if biases had a significant effect on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' However, SCHOOL UNIFORM is not significantly correlated with preventive behaviours, which is clearly different from the results of WASHING EDUCATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' This suggests, to a certain extent, the biases are minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Wearing masks are less effective in open air than indoors [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In contrast to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' [1], we used only three proxies for preventive behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Therefore, we did not scrutinise how hand washing and mask wearing changed in different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In contrast to hand washing, the benefit of mask wearing depends on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Wearing masks in the open air has limited effectiveness [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In mid-summer, wearing masks increased the risk of heatstroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' In this situation, the cost of wearing a mask is 12 higher than its benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' It is, therefore, important to examine mask-wearing behaviour in various situations in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Conclusion Preventive behaviours play a vital role in coping with unexpected pandemics such as COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' We concluded that people can form sustainable development-related habits through childhood hygiene practice education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Acknowledgements We would like to thank Editage (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='editage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content='com) for their English Language editing and reviewing of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Authors’ contributions EY and FO participated in the conceptualisation of the study and analysed the patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' YT designed the panel survey and performed data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' EY wrote the main text and made the tables for the original manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' All authors reviewed, edited, and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The authors are responsible for any errors in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Funding This study was supported by the Fostering Joint International Research B (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 18KK0048) and the Grant-in-Aid for Scientific Research S (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 20H05632) from the Japan Society for the Promotion of Science to Yoshiro Tsutsui and Fumio Ohtake, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Availability of data and materials The datasets used and analysed in this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Declarations 13 Ethics approval and consent to participate This study was conducted with the ex-ante approval of the Ethics Committee of the Graduate School of Economics, Osaka University, and all methods were carried out in accordance with the relevant guidelines and regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' The ethics approval number of Osaka University for this study is R021014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Informed consent for study participation was obtained from all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Author details 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Department of Economics, Seinan Gakuin University, Fukuoka, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FJT4oBgHgl3EQfMyyd/content/2301.11475v1.pdf'} +page_content=' Faculty of Social Relations, Kyoto Bunkyo University, Kyoto, JAPAN, 3.' metadata={'source': 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classifies +the tweets related to birthday posts into two +classes namely, exact age(positive class) and +non-exact age(negative class). We made two +submissions with variations in the preprocess- +ing of text which yielded F1 scores of 0.80 and +0.81 when evaluated by the organizers. +1 +Introduction +Determining the exact age of an individual is cru- +cial to increasing the use of social media data for +research purposes. In this contemporary world, +adolescents use social media to the extent that it +can have some very severe effects on their overall +well-being if not monitored. Hence, a few applica- +tions like Twitter have set up some age restrictions +for the well-being of an individual. They auto- +matically detect the age of an individual trying to +protect them from viewing unnecessary and harm- +ful content. +In this work, we determine the exact age of an +individual based on their tweets on Twitter. This +helps in validating if a particular user has faked +their age or not. One of the major challenges we +faced while working with the data is that some of +them could tweet about their friend’s or relative’s +birthday which was getting misclassified. We have +used BERT model which helps in the binary classi- +fication of our data. While developing our system +for this task, we have discovered BERT that outper- +forms traditional training models. +2 +Methodology +2.1 +Pre-Processing +Initially, the organisers provided us with 8,800 +training data and 2,200 validation data. This data- +set consisted of three fields: tweet id, text of +the Tweet Object and annotated binary class la- +bel(exact age present/absent). The training and +validation data were later combined and it was +pre-processed for further development. For pre- +processing, we removed URLs, emoticons, hash- +tags, and mentions using a python package tweet- +preprocessor. After that we removed: contrac- +tions from the tweets, special characters and ex- +tra spaces. Then we used python package called +Natural Language Toolkit for removing the stop +words. After these steps, we have further divided +the pre-processing into two techniques: pronouns +and removing pronouns. +2.2 +Model +In our model, we have used BERT (bert-base- +uncased) from the Hugging Face library as a clas- +sifier and Softmax as the activation function. In the +BERT model, there is an important special token +[CLS] which is used as an input for our choice of +classifier. We have used the Adam optimizer to fine- +tune our BERT model. We trained the model with +4 to 10 epochs which converged after 10 epochs. +Learning rate of the optimizer is given by 5e - 5.The +batch size used is 32. +3 +Evaluation +In the validation phase, our model produced satis- +factory results with about 90%. In the test data, +10,000 tweets were provided by the organizers. +We have first pre-processed with pronouns and +then removed pronouns in the next round of pre- +processing. After evaluation, our models generated +an F1-score of 0.80 and 0.81. +Table 1 shows our evaluation scores for Precision, +arXiv:2301.05395v1 [cs.CL] 12 Dec 2022 + +Model +Precision +Recall +F1-Score +Model 1 +0.839 +0.780 +0.808 +Model 2 +0.771 +0.870 +0.818 +Table 1: Evaluation scores +Recall, and F1-Score as provided by the organizers. +Model 1 shows scores of pre-processing with pro- +nouns and Model 2 shows scores of pre-processing +with pronouns removed. +4 +Conclusion +We discussed our approach to fine-tuning our BERT +model on Task 4 of the 2022 Social Media Min- +ing for Health applications shared task. As we +observe from the results, the given training data +was inadequate to train on a BERT model. There +was an imbalance in the number of positives and +negatives given in our dataset(refer to Figure 1). +An interesting observation drawn from this work +is that BERT models rely on huge and balanced +datasets for learning patterns. Future work might +consider collecting more data points for training, +fine-tuning our BERT model, and applying other +state-of-the-art methods like RoBERTa. +0 +1 +5,966 +2,834 +Figure 1: Summary of Dataset Labels +References +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and +Kristina Toutanova. 2018. +BERT: pre-training of +deep bidirectional transformers for language under- +standing. CoRR, abs/1810.04805. +Katikapalli Subramanyam Kalyan, Ajit Rajasekharan, +and Sivanesan Sangeetha. 2021. Ammus : A sur- +vey of transformer-based pretrained models in natu- +ral language processing. +Diederik P. Kingma and Jimmy Ba. 2014. Adam: A +method for stochastic optimization. +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- +dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, +Luke Zettlemoyer, and Veselin Stoyanov. 2019. +Roberta: A robustly optimized bert pretraining ap- +proach. +Arjun Magge, Ari Klein, Antonio Miranda-Escalada, +Mohammed Ali Al-garadi, Ilseyar Alimova, Zul- +fat +Miftahutdinov, +Eulalia +Farre-Maduell, +Sal- +vador Lima Lopez, Ivan Flores, Karen O’Connor, +Davy Weissenbacher, +Elena Tutubalina, +Abeed +Sarker, Juan M Banda, Martin Krallinger, and Gra- +ciela Gonzalez-Hernandez, editors. 2021. Proceed- +ings of the Sixth Social Media Mining for Health +(#SMM4H) Workshop and Shared Task. Association +for Computational Linguistics, Mexico City, Mex- +ico. + diff --git a/XdE5T4oBgHgl3EQfCg4x/content/tmp_files/load_file.txt b/XdE5T4oBgHgl3EQfCg4x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7a8816bd8814d7c67270795e9add06eb0ed5e78 --- /dev/null +++ b/XdE5T4oBgHgl3EQfCg4x/content/tmp_files/load_file.txt @@ -0,0 +1,74 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf,len=73 +page_content='MaNLP@SMM4H’22: BERT for Classification of Twitter Posts Keshav Kapur Manipal Institute of Technology keshav29kapur@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='com Rajitha Harikrishnan Manipal Institute of Technology rajithasuja@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='com Sanjay Singh Manipal Institute of Technology sanjay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='singh@manipal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='edu Abstract The reported work is our straightforward ap- proach for the shared task “Classification of tweets self-reporting age” organized by the “Social Media Mining for Health Applications (SMM4H)” workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' This literature de- scribes the approach that was used to build a binary classification system, that classifies the tweets related to birthday posts into two classes namely, exact age(positive class) and non-exact age(negative class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' We made two submissions with variations in the preprocess- ing of text which yielded F1 scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='80 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='81 when evaluated by the organizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 1 Introduction Determining the exact age of an individual is cru- cial to increasing the use of social media data for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' In this contemporary world, adolescents use social media to the extent that it can have some very severe effects on their overall well-being if not monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Hence, a few applica- tions like Twitter have set up some age restrictions for the well-being of an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' They auto- matically detect the age of an individual trying to protect them from viewing unnecessary and harm- ful content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' In this work, we determine the exact age of an individual based on their tweets on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' This helps in validating if a particular user has faked their age or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' One of the major challenges we faced while working with the data is that some of them could tweet about their friend’s or relative’s birthday which was getting misclassified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' We have used BERT model which helps in the binary classi- fication of our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' While developing our system for this task, we have discovered BERT that outper- forms traditional training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='1 Pre-Processing Initially, the organisers provided us with 8,800 training data and 2,200 validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' This data- set consisted of three fields: tweet id, text of the Tweet Object and annotated binary class la- bel(exact age present/absent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' The training and validation data were later combined and it was pre-processed for further development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' For pre- processing, we removed URLs, emoticons, hash- tags, and mentions using a python package tweet- preprocessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' After that we removed: contrac- tions from the tweets, special characters and ex- tra spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Then we used python package called Natural Language Toolkit for removing the stop words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' After these steps, we have further divided the pre-processing into two techniques: pronouns and removing pronouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='2 Model In our model, we have used BERT (bert-base- uncased) from the Hugging Face library as a clas- sifier and Softmax as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' In the BERT model, there is an important special token [CLS] which is used as an input for our choice of classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' We have used the Adam optimizer to fine- tune our BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' We trained the model with 4 to 10 epochs which converged after 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Learning rate of the optimizer is given by 5e - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='The batch size used is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 3 Evaluation In the validation phase, our model produced satis- factory results with about 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' In the test data, 10,000 tweets were provided by the organizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' We have first pre-processed with pronouns and then removed pronouns in the next round of pre- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' After evaluation, our models generated an F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='80 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Table 1 shows our evaluation scores for Precision, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='05395v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='CL] 12 Dec 2022 Model Precision Recall F1-Score Model 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='808 Model 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='818 Table 1: Evaluation scores Recall, and F1-Score as provided by the organizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Model 1 shows scores of pre-processing with pro- nouns and Model 2 shows scores of pre-processing with pronouns removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 4 Conclusion We discussed our approach to fine-tuning our BERT model on Task 4 of the 2022 Social Media Min- ing for Health applications shared task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' As we observe from the results, the given training data was inadequate to train on a BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' There was an imbalance in the number of positives and negatives given in our dataset(refer to Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' An interesting observation drawn from this work is that BERT models rely on huge and balanced datasets for learning patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Future work might consider collecting more data points for training, fine-tuning our BERT model, and applying other state-of-the-art methods like RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 0 1 5,966 2,834 Figure 1: Summary of Dataset Labels References Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' BERT: pre-training of deep bidirectional transformers for language under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' CoRR, abs/1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content='04805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Katikapalli Subramanyam Kalyan, Ajit Rajasekharan, and Sivanesan Sangeetha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Ammus : A sur- vey of transformer-based pretrained models in natu- ral language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Roberta: A robustly optimized bert pretraining ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Arjun Magge, Ari Klein, Antonio Miranda-Escalada, Mohammed Ali Al-garadi, Ilseyar Alimova, Zul- fat Miftahutdinov, Eulalia Farre-Maduell, Sal- vador Lima Lopez, Ivan Flores, Karen O’Connor, Davy Weissenbacher, Elena Tutubalina, Abeed Sarker, Juan M Banda, Martin Krallinger, and Gra- ciela Gonzalez-Hernandez, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Proceed- ings of the Sixth Social Media Mining for Health (#SMM4H) Workshop and Shared Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} +page_content=' Association for Computational Linguistics, Mexico City, Mex- ico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE5T4oBgHgl3EQfCg4x/content/2301.05395v1.pdf'} diff --git a/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/2301.01811v1.pdf.txt b/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/2301.01811v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5adefc3f1356e6097b6131fe0174e3812d1de3df --- /dev/null +++ b/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/2301.01811v1.pdf.txt @@ -0,0 +1,1495 @@ +On using Reproducible Hilbert Spaces for the +analysis of Replicated Spatial Point Processes. +A. Sim´o +Department of Mathematics-IMAC. Universitat Jaume I. Avda. del Riu Sec s/n. 12071-Castell´o, Spain. +January 6, 2023 +Abstract +This paper focuses on the use of the theory of Reproducing Kernel +Hilbert Spaces in the statistical analysis of replicated point processes. We +show that spatial point processes can be observed as random variables in +a Reproducing Kernel Hilbert Space and, as a result, methodological and +theoretical results for statistical analysis in these spaces can be applied to +them. In particular and by way of illustration, we show how we can use +the proposed methodology to identify differences between several classes +of replicate point patterns using the MBox and MANOVA tests, and to +classify a new observation, using Discriminant Functions. +keyword +Reproducing Kernel Hilbert Space; Functional Data Analysis; Repli- +cated spatial point processes; Analysis of Variance; Supervised Classifica- +tion +1 +Introduction +A spatial point process is a stochastic random process whose realizations are +locally finite sets of points (locations of events) in a study region E of R2. The +term locally finite means that for any Borel bounded set there is a finite number +of points (with probability one). +Spatial point patterns arise as the natural +sampling information in many problems. Examples include the positions of trees +in a forest, galaxies in the sky, a certain type of commerce in a city or cases +of a certain disease. Seminal books on the theory of point processes and their +applications include Stoyan et al. (1988); Stoyan and Stoyan (1994); Baddeley +et al. (2007); Illian et al. (2008); Diggle (2013); Cressie (2015). +In mathematical terms, if (E)n ≡ E • E • · · · • E (n times) is the set of n +elements of E; +We define the exponential space as +Ee ≡ +∞ +� +n=0 +(E)n. +1 +arXiv:2301.01811v1 [stat.ME] 4 Jan 2023 + +A point process, Φ, is defined as a measurable application of a probability space +(Ω, A, P′) on the measurable space (Ee, Be). +For details on the measurable +exponential space see Carter and Prenter (1972). +Accurate and well-founded methodologies for the analysis of point processes +are widely used in the literature, but most of them focus on the case where only +one observation of the process is available. +There are mainly two different approaches to represent and/or describe point +processes: event densities and distributions and random counting measures. In +this paper we focus on the second one. Our goal is to take advantage of the +relationship between random measures and RKHS-valued random variables, to +work with point processes characterized by random functions in a RKHS. Thus, +we explore how we can deal with more complex statistical methodologies in this +space. In particular, our proposals will make it very natural and straightforward +to analyze replicated point patterns, i.e., data sets consisting of several point +patterns that can be considered independent replicates of the same experiment. +RKHS spaces are well known in the statistical literature, and are frequently +used in the context of Machine Learning (Berlinet and Thomas-Agnan, 2011; +Saitoh and Sawano, 2016) as a valuable tool in classification or regression prob- +lems on Euclidean or L2 spaces or even on Riemannian manifolds. In particular, +in Euclidean spaces the success of many classification algorithms is due to the +use of kernel methods (Sch¨olkopf et al., 2002). However, there is hardly any lit- +erature on statistical methods when the data live in a RKHS (Luki´c and Beder, +2001), which is our case. +The theory of statistics with functional data is an important field of research +in statistics. It deals with samples in which a function is observed for each indi- +vidual. The books by Silverman and Ramsay (2005); Ferraty and Vieu (2006); +Horv´ath and Kokoszka (2012) and Aneiros et al. (2017) are key references, as are +the excellent reviews by Cuevas (2014) and Goia and Vieu (2016). Although the +theory of functional data analysis has incorporated many tools from classical +parametric or nonparametric statistics, the infinite-dimensional nature of the +sample space poses particular problems (Ferraty and Vieu, 2006), even though, +in practice, one has only sampled observed curves into a finite set of observation +points. +There are two different perspectives on functional data. The first view is that +functional data are realizations of random variables taking values in a Hilbert +space. The second view is that functional data are the sample trajectories of a +stochastic process with smooth mean and covariance functions. There are subtle +differences between the two perspectives from a theoretical point of view. +RKHSs are often present in the second perspective of functional data analysis +(Preda, 2007; Kadri et al., 2016) since the Lo´eve-Parzen congruence (Aronszajn, +1950) links a second-order stochastic process with the RKHS generated by its +covariance function (Eubank and Hsing, 2008; Kupresanin et al., 2010). The +functional principal component directions turn out to be an orthonormal ba- +sis of the Hilbert-Schmidt covariance operator associated with the covariance +2 + +kernel (Horv´ath and Kokoszka, 2012; Cuevas, 2014). Although, in our case the +functional data be by definition a sample of a variable in a Hilbert space (first +point of view), our approximation will be similar to the latter. +In Baddeley (2015), the practical analysis of replicated point patterns is ex- +plained using the R package Spatstat. In particular, one of the dataset included +in this package will be used in this paper: the Pyramidal dataset. This dataset +contains data from Diggle et al. (1991) and they are locations of pyramidal +neurons in human brain of 12 normal, 9 schizoaffective, and 10 schizophrenic +human subjects. All our implementations were written with R (R Core Team, +2021), mainly using the Spatstat and the MASS packages. +The article is organized as follows: +First at all, Section 2 concerns the theoretical concepts. Secondly, in Section 3 +the theoretical concepts are applied to the statistical analysis of replicated point +patterns. After taht, two particular applications are detailed in Section 4. Fi- +nally, conclusions are discussed in Section 5. +2 +From point processes to random elements in +a reproducing kernel Hilbert space +As indicated in the introduction, the objective of this paper is to show a method- +ology that allows the study of point processes by characterizing them by means +of random functions in a RKHS. In this section we introduce the theoretical +concepts necessary for this purpose. First, the definition and properties of re- +producible kernel Hilbert spaces are briefly introduced. +Second, we see how +to embed measures in a RKHS. Thereafter, we express a point process as a +random measure and discuss some theoretical results on random variables in +Hilbert spaces, in general, and RKHS, in particular. +2.1 +Reproducible kernel Hilbert spaces +A Reproducible Kernel Hilbert Space (RKHS) is a Hilbert space of functions +f : E → R with some practical and interesting properties. +The theory of +Reproducible Kernel Hilbert Spaces was developed by Aronszajn (1950). +Definition 1. Let H be a Hilbert space of real-valued functions defined on E +and ⟨·, ·⟩H the inner product on H. A function k : E × Rn → R, is said to be +an reproducing kernel (rk) associated with H if it satisfies: +1. for every x ∈ E, k(·, x) ∈ H. +2. k satisfies the ”reproducing property”; that is, ∀f ∈ H and x ∈ E +f(x) = ⟨k(·, x), f⟩H +Definition 2. A Hilbert space of real-valued functions is a Reproducible Kernel +Hilbert Space if it has a reproducing kernel (rk) associated. +3 + +A RKHS can be obtained from a kernel and each kernel determines (Moore- +Aronszajn theorem) a unique RKHS, denoted by Hk. The construction of Hk +is given as follows. +We consider the set H0 of linear combinations: +H0 = { +N +� +i=1 +bik(·, xi), N ≥ 1, bi ∈ R, xi ∈ E}, +the RKHS H associated with the kernel k is the closure of H0. +Given φ1 = �N1 +i=1 aik(·, xi) and φ2 = �N2 +i=1 bik(·, yi), ⟨φ1, φ2⟩k = � +i +� +j aibjk(xi, yj) +2.2 +Embedding measures in a RKHS +The study of random measures requires sophisticated mathematical tools. For +this reason, and following Berlinet and Thomas-Agnan (2011), we first show +how reproducing kernels can be used to represent measures in functional spaces +starting with Dirac measures. We will then use this embedding to define inner +products. +Given a compact subset E of R2, and B the σ−algebra of Borel of subsets +of E, the Dirac measure δx is defined for x in E by: +δx(A) = +� 1 +if +x ∈ A +0 +if +x /∈ A +where A is a Borel set in E. +The mapping: +δx �→ k(·, x) +embeds the set of Dirac measures on E in the RKHS with kernel k. +If the function k(y, ·) is measurable, the value k(y, x) of the function k(·, x) +at the point y can be written as the integral +� +k(y, t)dδx(t), and the mapping +can be rewritten as: +δx �→ Iδx = +� +k(y, t)dδx(t) +In addition, for any measurable function f in Hk ⟨f, k(·, x)⟩H = f(x) = +� +f(t)dδx(t) +More generally, if x1, ..., xN are N distinct points in E and b1, ..., bN, N non +null real numbers, a linear combination +µ = +N +� +i=1 +biδxi +of Dirac measures is called finite support signed measure. We can extend the +previous mapping with +µ �→ +N +� +i=1 +bik(·, xi) = +� +k(y, t)dµ(t). +(1) +4 + +This mapping embeds in Hk the set of measures on E with finite support, +M0, and the set H0 can be seen as the set of its representers in Hk. Again, we +have the property that for any measurable function f in Hk ⟨f, �N +i=1 bik(·, xi)⟩H = +�N +i=1 f(xi) = +� +fdµ. +Following the generalisation, we can also embed the set M of signed measures +on E in Hk with the mapping: +M +→ +Hk +µ +�→ +Iµ = +� +k(·, t)dµ(t) +see Berlinet and Thomas-Agnan (2011) to more details. +If we assume that k is such that the functions Iµ and Iν are different if µ and +ν are not equal, Theorem 99 of Berlinet and Thomas-Agnan (2011) guaranties +that the mapping: +M × M +→ +R +(µ, ν) +�→ +⟨Iµ, Iν⟩H +defines an inner product on M for which M0 is dense in M and its converse. +Guilbart (1979) was the pioneer in studying the relationships between repro- +ducing kernels and inner products on the space M. He exploited the embedding +and characterized the inner products inducing the weak topology on sets of mea- +sures. +2.3 +Point processes and random measures +If Φ is a point process, the random counting measure associated to Φ is defined +as: +µΦ(B) = #{x : x ∈ Φ ∩ B}, +(2) +for B a Borel set in E. +The random counting measure of a point process characterizes its probabil- +ity distribution and provides a framework for developing the theory of point +processes as part of a general theory of random measures Daley and Vere-Jones +(2008). +If M is a set of measures in E equipped with some σ-algebra, a random +measure can be regarded as a random variable with values in M. For a detailed +study of the theory of random measures, see Daley and Vere-Jones (1998). +Although this concept seems easy, the very definition of random measures raises +delicate problems. For instance, the definition of the σ-algebra possibly derived +from some topology on M is not a simple matter and the resulting theory +involves delicate mathematical questions. +For this reason, once we have seen how we can embed the set of measures in a +RKHS, we use this embedding to define and study random measures as random +elements in a RKHS i.e. RKHS-valued random variables. We will assume that +5 + +the random variable takes its values in M (or M0) with probability 1. This is +the construction proposed by (Suquet, 1986). +With this construction, we can subsequently exploit the known results con- +cerning the probability laws in a separable Hilbert space. In addition, RKHS +are vectorial metric spaces and they can be considered as the natural extension +of the usual Euclidean spaces. Most of the theoretical and methodological sta- +tistical results defined in Euclidean spaces are directly inhered in RKHS spaces. +Furthermore, the completeness of Hilbert spaces gives a framework in which to +work with infinite-dimensional vectors as the limit of finite-dimensional vectors. +Probability theory in Banach and Hilbert spaces is an important branch +of modern probability. +A complete treatment of this topic can be found in +Ledoux and Talagrand (1991) and Hsing and Eubank (2015). A large number +of concerning large sample results can be applied on a separable Hilbert space. +Among all these theoretical results, we recall here only a central limit theorem +which guarantees convergence to a Gaussian process analogous to Euclidean +spaces. See Hsing and Eubank (2015) for a complete exposition. +Theorem 3. Let χ1, ..., χn be independent and identically distributed random +elements in a Hilbert space H with mean 0 and E∥χi∥2 < ∞. Then +ξn := +�n +i=1 χi +� +(n) +→d ξ, +where ξ is Gaussian random element of H with covariance operator equal to +E(χi ⊗ χi). Being ⊗ the tensor product operator in H, ∥ · ∥ the norm defined by +the interior product in H and →d denotes convergence in distribution. +Furthermore, if the Hilbert space is also a RKHS, we have more important +and strong properties which guarantees the application of standard statistical +techniques to random variables in a RKHS. Guilbart (1979) proved a Glivenko- +Cantelli theorem that he applied to estimation and hypothesis testing. Berlinet +(1980b,a) studied weak convergence in the set of probabilities on a RKHS, mea- +surability and integrability of RKHS-valued variables. Another important re- +sult, although it will not be used in this paper, is the theorem 7.5.1 of (Hsing +and Eubank, 2015), which tells us that a random variable in a RKHS is also a +stochastic process and the reciprocal. +Turning to the case of point processes, Equation 2 can be rewritten as: +µΦ(B) = +� +xi∈Φ +δxi(B) = +N +� +i +δxi(B), +(3) +for B a Borel set in E and where δ denotes the Dirac measure. +The counting measure defined from Φ is a random measure and all previously +mentioned for random measures applies to the statistical analysis of replicated +point processes. +6 + +A counting measure is a particular case of finite support measure and we +can embed it in a RKHS using the mapping 1: +µΦ �→ +N +� +i=1 +k(·, xi). +(4) +And, we will assume that its associated RKHS random variable takes its values +in the set of counting measures on E, N ⊂ M0, with probability 1. +Moreover, it is satisfied that if two counting measures µ and ν are not equal, +i.e., they have different supports {x1, ..., xN} and {y1, ..., yM}, Iµ = �N +i=1 k(·, xi) +and Iν = �M +i=1 k(·, yi) are different. +Regarding computational aspects, the mapping 4 can be easily obtained +using, for example, the function density of the spatstat package of R. +Although the possibilities opened up by this way of working are enormous, +in this paper, we just focus on two applications for illustrative purposes. +In the first application, we have different experimental groups, we observe +independent replicates of a point process within each group and we are interested +in contrasting whether there are differences between groups, i.e. an ANOVA +problem where the response is a point pattern. In the second example, we also +have different groups but we are interested in a rule to classify a new point +pattern in one of these groups, i.e. a supervised classification problem when the +explanatory variable is a point pattern. +In both cases, classical multivariate statistical methods will be used in order +to take profit of all the theoretical results aforementioned. +3 +Application to the statistical analysis of repli- +cated point processes +In the previous section, we have seen how to transform point processes into +random elements in a RKHS ; therefore, in this section, we assume that we +have a sample {ϕi(·)}n +i=1 of a random element in the RKHS Hk, each of the +elements of the sample having the expression: +ϕi(x) = +Ni +� +l=1 +k(x, xil), x ∈ E +(5) +Because our data are a particular kind of functions, it would make sense to +use the insights of functional data analysis (FDA) to carry out any statistical +analysis. But, our data are very different to the typical data in FDA issues. +Firstly, our data are not expressed in the standard form, where the i-th func- +tional datum is given just by a set of discrete measured values ((yi1, t1), · · · , (yimi, tmi)) +with little knowledge of the analytical form of the function. Each datum of our +sample is a function ϕi(x) whose analytical expression is defined by (Eq. 5). +7 + +Secondly, our function space is a RKHS, i.e. our raw functional data “live” in +a RKHS and, as it was said before, its vector structure and inner product can +be exploited in the data analysis. For these reasons, our proposal is to express +each function of our dataset with respect to the orthonormal base given by the +eigenfunction decomposition of the kernel that defines the RKHS. The elements +given by the eigenfunction decomposition of the kernel are orthonormal and are +ordered according to an optimality approximation criterion. These properties +allow us to reduce the dimension and thus we can apply classical statistical +procedures as in the multivariate Euclidean case. Similar ideas were previously +used by authors in a very different context: supervising classification of geomet- +rical object. The following results are similar to results of Section 3 in Barahona +et al. (2018). +To obtain the projection on this base, we need first to change the expression +of our functions so that the points at which the kernel is evaluated are the +same in all the point patterns of the sample. For that we can use the following +theorem. +Theorem 4 (Representer Theorem). Given ϕi(·), a grid {al}N +l=1 in E, denoting +ϕi(al) = bil and given a regularization parameter γ > 0 then ∃!χi : R2 → R; +χi(y) = �N +l=1 βilk(y, al) such as: +χi = arg min +g∈Hk +1 +N +N +� +i=1 +(g(al) − bil)2 + γ ∥g∥2 +Hk , +(6) +where βil ∈ R (for l = 1, . . . , N) are the solutions of: +(γ N IN×N + K|a)βi = bi, +with K|a the matrix defined as (K|a)(i, j) = k(ai, aj), i, j = 1, . . . , N, and βi, +bi are the N × 1 vectors +βi = +� +� +� +� +� +βi1 +βi2 +... +βiN +� +� +� +� +� ; +bi = +� +� +� +� +� +bi1 +bi2 +... +biN +� +� +� +� +� +As a result of applying the theorem, from now on we will work with the +sample: +{χi(·) = +N +� +l=1 +βilk(·, al)}n +i=1 +(7) +It is well known (Hsing and Eubank, 2015) that if E is compact and k +continuous, measurable and bounded, the integral operator associated to the +kernel function k and defined by: +(Kf)(·) = +� +E +k(·, x)f(x)dx, +f ∈ L2(E), +(8) +8 + +where L2(E) is the space of square integrable functions on E, is a compact, +continuous, self-adjoint, and positive operator. +As a result, it can be expressed as +K = +� +q≥1 +λqeq ⊗ eq, +where {λq, eq}q is the countable sequence of its eigenvalues and (orthonormal) +eigenfunctions (spectral decomposition). +In addition: +k(x, y) = +� +q≥1 +λqeq(x)eq(y) +. +And { +� +λqeq}q is a complete orthonormal basis for Hk. +The following results show us how we can project our sample in this base of +the RKHS. +Theorem 5. Let χi be an element of the RKHS Hk, it can be expressed as +χi(·) = +∞ +� +q=1 +µqi +�� +λqeq(·) +� +, +(9) +where { +� +λqeq}∞ +q=1. +In addition, if χi(·) = +∞ +� +q=1 +µqi +�� +λqeq(·) +� +and χj(·) = +∞ +� +q=1 +µqj +�� +λqeq(·) +� +, +the inner product is: +⟨ϕi, ϕj⟩k = +� +q +µqiµqj. +Furthermore (Smale and Zhou, 2009; Gonz´alez and Mu˜noz, 2010), the first +d = rank(K|a) coefficients µqi can be approximated by +� +µqi = +� +ℓq(vq · βi) +(10) +where vq ∈ RN are the eigenvectors of K|a, ℓq are the eigenvalues of K|a. +The following result assures us that if we express our infinite-dimensional +data in this basis and truncate it to obtain a finite-dimensional vector sample, +we will have the highest possible accuracy. +Proposition 6. Theorems 4.4.7 and 4.6.8 in Hsing and Eubank (2015) tell us +that for a fixed integer r > 0 with λr > 0: +min +f1,...,fr∈Hk +� � +E×E +� +k(y, x) − +r +� +q=1 +fq(y)fq(x) +�2 +dydx, +the minimum is achieved by fq = eq. +9 + +This result ensures that the truncated eigenvalue-eigenvector decomposition +provides the best approximation to k and, as a result, to our data (Eq. 7). +If we truncate the summation in Equation 9 to a low number of terms r, +r ≤ d = rank(K|a), each point pattern ϕi for i = 1, . . . , n, is given by the +coefficients µqi for q = 1, . . . , r (estimated by � +µqi), on the orthonormal basis +{ +� +λqψq}∞ +q=1. As a result, it can be represented as the r-dimensional vector +µi = ( � +µ1i, � +µ2i, . . . , � +µri). +(11) +This expression optimally reduces the infinite-dimensional problem to a +finite-dimensional problem. +It is now possible to apply well-known classical +multivariate methods. In particular, in the following section two well-known +classical multivariate methods will be used for illustrative purposes: MANOVA +and Discriminant Analysis. +4 +Application to locations of pyramidal neurons +in human brain +As mentioned before, the Pyramidal dataset is included in the R package spastat +and contains data from Diggle et al. (1991). +The data are the locations of +pyramidal neurons in human brain. One point pattern was observed in each +of 31 human subjects. There were 12 normal control, 9 schizoaffective, and 10 +schizophrenic cases. See Diggle et al. (1991) for a more detailed explanation of +them. Figure 1 gives plots of the point patterns in our sample, one plot for each +subject. +In this example E = [0, 1] × [0, 1] and a gaussian kernel with σ = 0.05 were +used to embed the point patterns in a RKHS using the map 4. A grid {al}N +l=1 +with al equally spaced in E with a vertical and horizontal step of h = 0.02 is used +to apply the representer theorem 4. The parameter γ = 0.000127 in Equation 6 +is fixed to the minimum value wich makes matrix (γ N IN×N + K|a) invertible. +The parameters h and σ are related with the accuracy of our working data. +When h and σ decrease, we have more accuracy, but more computational cost +and complications in the calculations. The chosen values represent a balance +between both. +After applying the representer theorem we have a sample {χi(·), yi}31 +i=1 with +χi(·) = �N +l=1 βi,lk(·, al), and yi a categorical variable indicating if the subject +is normal, schizoaffective or schizophrenic. +In Figure 2 we can see the plot of one point pattern of each group and its +corresponding elements ϕi and χi in the RKHS. +Finally, Equation 9 is applied to obtain the vector of coefficients µi that +represent each point pattern in relation to the orthonormal base of Hk. Because +the size of our sample is very small (12, 9 and 10 cases in each group), we +truncate it at r = 6. +It is worth analysing the meaning of the firsts functions of the base. +In +Figure 3, we can see the six firsts eigen functions. The coefficient of the first +10 + +Figure 1: Positions of the pyramidal neurons for the three groups of subjects. +11 + +Normal +1 +2 +3 +4 +o o +.90 +8. +00 +o0 +0 +00 +8 +.8 +0 +0 +80. +8.8 +o +8 +0 +18 +0 +.o 0 +. +8. +20 +00 +Pe +8 +0 +5 +6 +7 +8 + o +0 +2.08.0 +00 +0o +0 +o0 +0 +c00.,0 +0 +.80 +00 +0 +00 +0 +T8 +o° +88 +0 +0 +0 +6 +10 +11 +12 +0 +0o +0 +o0 +0 +0 +0 +00 +0 +0%0 +.°。0 +0 +0 +00 +0 +° +08080 +.88 +608 +80 +0 0 +oo +8 +0.: +0 +0 +0Schizoaffective +13 +14 +15 +9 0 +0 0 +0 +0 +00 +0 +o0 +0 +0 +16 +17 +18 +.0 +0 +0 +00 +o 0 +00 +18 +960.000 +30 +19 +20 +21 +0 +0 +o +0 +00 +og +0 +0 +80 +0. +00 +0 +0 +.8 +%.° +0. +0 0 +0 oo +00%0 +0 +8 +08 +eSchizophrenic +22 +23 +24 +25 +68 ++00 0 0 +0 +00 +0 +%00 +0 +0 +000 +20. +90 +8 +26 +27 +28 +29 +.% +004 +0o9 +800 +000 +;0 +8 +0 +.%0 00 +0 +0 0 +80 +0 +00 +00 +8 +0 0 +30 +31 +00 +0 +% +0 +o% +0function measure high density concentrated in the center of the window, the +second and third high density just within one half of the window and so on. +4.1 +ANOVA +Two principal approaches can be found in the literature of spatial point pat- +terns to identify significant differences between several experimental groups +(Diggle et al., 2000). The first one (Diggle et al., 1991; Ram´on et al., 2016; +Gonz´alez Monsalve et al., 2018) is based on functional descriptors of the pat- +tern and nonparametric inference. +The second one is based on assuming a parametric model for the pattern +(usually pairwise interaction point process or Gibbs processes), the parame- +ter of the models are estimated using maximum likelihood or pseudo-likelihood +methods for each group and differences between groups are tested by compar- +ing fits with and without the assumption of common parameter (Illian and +Hendrichsen, 2010; Illian et al., 2012). +Following the aforementioned procedure, we will work with the RKHS ele- +ments resulting of the embedding. +Since a RKHS is a vector space, the mean sample element is simply obtained +as: ¯χ(·) = �N +l=1 +�n +i +βi,l +n k(·, al). In Figure 4 we can see the mean sample element +of each group. +In this example, we are only focused on the answer to the following question: +do the observed patterns differ significantly in mean from group to group? For +this purpose, a Multivariate ANOVA test can be applied to the r-variate sample +{µi, yi}31 +i=1. Before the MANOVA test, a Box’s M test must be applied to test +the assumption of homogeneity of variance-covariance matrices. +The results +of both tests can be found in Table 1. +At a significance level of 0.05, the +hypothesis of homogeneity of variance-covariance matrices would be accepted +and the hypothesis of equality of means would be rejected. +It is important to note that, in this case, the sample size is small and there- +fore, the Central Limit Theorem is not applicable to assume normality. Shapiro +univariate tests applied to the residuals gave non significative results. Although +these results do not guarantee multivariate normality, is not a cause for con- +cern for two reasons. The first reason is the robustness to non-normality of the +MANOVA test, and the second is that this example is only illustrative. +Once the multivariate hypothesis of equality of means has been rejected, we +perform an univariate ANOVA test to analyse with more detail the differences. +The respective p-values for the first six coefficients of our base were: 0.069, +0.044, 0.066, 0.42, 0.58 and 0.115, and we can conclude that the differences +between classes are mainly in the four coefficients corresponding to the four +first functions of Figure 3. +It is important to emphasize again that, our objective in this section is to +show the potential applications of the proposed methodology, a deeper study +conducted in conjunction with experts in neuroanatomy would be necessary to +reach clinical conclusions. +12 + +Figure 2: Point pattern and RKHS functions of the first subject of each class +before and after applying the representer theorem. +Test +Statistic +df +p-value +Box’s M-test +36.009 +42 +0.7304 +Manova +2.2358 +12-48 +0.02451 +Table 1: Box’s M and MANOVA results. +13 + +0.8 +0.6 +0.4 +0.2 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.06 +80 +6 +0.4 +0.2 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.000 +6 +2 +0 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.00 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +p +0 +0 +0 +0 +00 +0 +0 +0 +0 +0 +00 +0 +0 +0 +00 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +P +0 +0 +0 +0 +0 +0 +0 +0 +0 +00 +0 +0 +00 +0 +0 +00 +0 +0 +0 +00 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +00 +0 +O0 +80 +0 +0.6 +0.4 +0.2 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.06 +0.4 +0.2 +0'0 +00 +0.2 +04 +06 +08 +1000. +0.6 +0 +2 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Figure 3: Firsts eigen functions of the Gaussian Kernel in a [0, 1]×[0, 1] window. +14 + +1.0 +1.0 +1.0 +8'0 +8'0 +0.6 +9'0 +0.4 +t0 +0.4 +0'0 +0'0 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0 +1.0 +1.0 +8'0 +8'0 +8'0 +8'0 +0.4 +0.4 +t0 +t0 +0.2 +0.2 +0'0 +0'0 +0'0 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.0 +80 +8'0 +8'0 +0.6 +9'0 +9'0 +9'0 +0.2 +0.2 +0.2 +0.2 +0'0 +0'0 +0'0 +00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Figure 4: The means of the RKHS functions of each class: normal, schizoaffec- +tive and schizophrenic. +4.2 +Discriminant Analysis +Once significant differences between groups of subjects have been found, it could +be very useful to find a decision rule to classify a new individual as normal, +schizoaffective or schizophrenic on the basis of its spatial pattern. +To our knowledge, the literature about supervised classification methods +applied to replicated point patterns is scarce and relies mainly on dissimilarity- +based methods (Mateu et al., 2015; Pawlasov´a and Dvoˇr´ak, 2022). +Similar +methods have been used for supervised classification of germ-grain models in +Gallego et al. (2016). +As it is well known, the literature on supervised classification methods is +extensive and covers a large number of methods ranging, from those based on +multivariate statistics to the most modern deep learning techniques (Hastie +et al., 2020). But, as mentioned before, in this paper, we focus only on classical +multivariate statistical methods, since the theorem 3 would allow us to take ad- +vantage of probability parametric models as well as convergence theorems. For +this reason, linear and quadratic discriminant functions (Venables and Ripley, +2013) will be used to illustrate the new methodology proposed in this work. As +it is known, both are parametric Bayesian methods that assume a multivari- +ate Gaussian model for the explanatory variables, with and without equality +of variance-covariance matrices respectively. The a priori probabilities of each +class will be estimated from the sample. +Since our dataset is very limited, we will first present some illustrations using +simulated point patterns. Two different experiments were performed. +In the first experiment two samples of size 20 of an homogeneous Poisson +point process (HPPP) with different intensities, λ1 and λ2, were simulated in a +rectangular window of size one. This experiment was repeated two times with +λ1 = 50 and λ1 = 90, respectively and λ2 = 100 in both cases. In Figure 5, +we can see one point pattern of each class and its corresponding element in the +RKHS. Since the difference between the two classes of point processes is in the +15 + +00 +0.6 +2 +0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.00.8 +0.6 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.08'0 +6 +? +0.0 +0.2 +0.4 +0.6 +0.8 +1.0mean of its counting measures, linear discriminant functions have been used in +both cases (lda function of R package MASS). The Table 2 shows the training +errors and the cross-validation errors, excellent results have been obtained, even +in the second case in which the difference between both models is very slight. +In the second experiment, two samples of size 30 of two different point pro- +cesses were simulated again in a [0, 1] × [0, 1] window. The first sample cor- +responds to a homogeneous Poisson point process with intensity λ1 = 36 and +the second to a Poisson cluster point process (PCPP) with the intensity of the +Poisson process of centres κ = 6, and each cluster consisting of 6 points in a +disc of radius 0.2. The resulting intensity is also λ2 = 36. In Figure 5, we can +see again one point pattern of each class and its corresponding element in the +RKHS. In this case, both point processes have the same intensity and the differ- +ence between both classes is given by the spatial variability, for this reason the +linear discriminant function does not work well and a quadratic discriminant +function must be used (qda function of R package MASS). +Figure 5: One simulated point patterns and its corresponding RKHS element of +each class. First row: homogeneous Poisson point processes with λ1 = 50 and +λ2 = 100, second row: homogeneous Poisson point processes with λ1 = 90 and +λ2 = 100. Third row: homogeneous Poisson point process with λ1 = 36 and +Poisson cluster point process with λ2 = 36. +16 + +0 +00 +0 +80 +0 +0 +00 +00 +8.0 +.0 +00 +CPO +8 +0 +% +8°00 +0 +0 +0.4 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.00.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.000000 +() +000 +8 +0 +98 +boo +00 +88 +8 +0 +00 +8 +00 +0 +0 +81.0 +00 +0 +6 +0 +0.4 +2 +0 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.00 +80 +0 +6 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.00 +0 +0 +0 +0 +0 +000 +0 +0 +08一 +00 +0.4 +0'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.000 +4. +0 +2 +0 +0 +00 +0.2 +04 +06 +08 +10The same accuracy parameters of the previous section have been used, i.e. +σ = 0.02 and h = 0.05. Regarding to the number of variables, i.e. the value of +r, different values ranging from r=6 to r=10 have been tested, but no major +differences have been found. In general, the best results were obtained for r=7, +and these are reported in Table 2. The results are again excellent. In addition +to obtaining very small cross-validation errors, the a posteriori probabilities of +well-classified cases are all practically greater than 0.95 and those of misclassified +cases lower than 0.6. +Models +Intensisties +Training error +CV error +HPPP-HPPP +λ1 = 50, λ2 = 100 +0 +0 +HPPP-HPPP +λ1 = 90, λ2 = 100 +0.1 +0.1755 +HPPP-PCPP +λ1 = 36, λ2 = 36 +0.05 +0.117 +Table 2: Supervised classification errors in the simulated experiments. +After testing the performance of the proposed methodology on a super- +vised classification problem, the lda function was applied to our real example +to find a decision rule to classify a new individual as normal, schizoaffective or +schizophrenic. We used linear discriminant analysis because the Box’s M test, +used to test the hypothesis of homogeneity of variances in the previous section, +did not yield significant differences. As expected, no good results were obtained +due to the limitations of the sample. The training error was relatively small +(0.29), but the cv error was not at all satisfactory (0.6). A larger dataset would +be necessary in order to be able to use in clinical practice. In our view, our +methodology could be used without modifications. +5 +Conclusions +We have introduced a new methodology for the statistical analysis of replicated +spatial point patterns. This methodology is based on the fact that the proba- +bility distribution of a point process is completely determined by its associated +random counting measure. Random measures can be embedded in a RKHS and, +in this way, we transform the point process in a random element in a RKHS, +where theoretical founded methods and algorithms can be applied, similar to +what is done in an Euclidean space. To do so, we express our data in the base +given by the kernel’s eigenfunctions and truncate this expression in the required +dimension. This guarantees to move to a lower dimension with the least loss of +accuracy. +As an example of the potential real-life applications of the proposed method- +ology, we have used it to detect differences between point patterns of pyramidal +neuron locations in the human brain from three groups of subjects (Diggle et al. +(1991). We have also used it to classify new observations using several simu- +lated datasets. With the results of these experiments, it can be stated that our +methodology is feasible for applications. +17 + +References +Aneiros, G., Bongiorno, E.G., Cao, R., Vieu, P., et al., 2017. Functional statis- +tics and related fields. Springer. +Aronszajn, N., 1950. Theory of reproducing kernels. Transactions of the Amer- +ican mathematical society 68, 337–404. +Baddeley, A., 2015. +Analysing replicated point patterns in spatstat. +Cran +Vignettes 35, 38. +Baddeley, A., B´ar´any, I., Schneider, R., 2007. Spatial point processes and their +applications. 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A toolbox for fitting complex spatial +point process models using integrated nested laplace approximation (inla). +The annals of applied statistics 6, 1499–1530. +Kadri, H., Duflos, E., Preux, P., Canu, S., Rakotomamonjy, A., Audiffren, J., +2016. Operator-valued kernels for learning from functional response data. The +Journal of Machine Learning Research 17, 613–666. +19 + +Kupresanin, A., Shin, H., King, D., Eubank, R., 2010. An rkhs framework for +functional data analysis. Journal of Statistical Planning and Inference 140, +3627–3637. +Ledoux, M., Talagrand, M., 1991. Probability in Banach Spaces: isoperimetry +and processes. volume 23. Springer Science & Business Media. +Luki´c, M., Beder, J., 2001. Stochastic processes with sample paths in reproduc- +ing kernel hilbert spaces. Transactions of the American Mathematical Society +353, 3945–3969. +Mateu, J., Schoenberg, F.P., Diez, D.M., Gonz´alez, J.A., Lu, W., 2015. On +measures of dissimilarity between point patterns: Classification based on pro- +totypes and multidimensional scaling. Biometrical Journal 57, 340–358. +Pawlasov´a, K., Dvoˇr´ak, J., 2022. Supervised nonparametric classification in the +context of replicated point patterns. Image Analysis & Stereology 41, 57–109. +Preda, C., 2007. Regression models for functional data by reproducing kernel +hilbert spaces methods. +Journal of statistical planning and inference 137, +829–840. +R Core Team, 2021. R: A Language and Environment for Statistical Computing. +R Foundation for Statistical Computing. Vienna, Austria. URL: https:// +www.R-project.org/. +Ram´on, P., de la Cruz, M., Chac´on-Labella, J., Escudero, A., 2016. A new +non-parametric method for analyzing replicated point patterns in ecology. +Ecography 39, 1109–1117. +Saitoh, S., Sawano, Y., 2016. Theory of reproducing kernels and applications. +Springer. +Sch¨olkopf, B., Smola, A.J., Bach, F., et al., 2002. Learning with kernels: support +vector machines, regularization, optimization, and beyond. MIT press. +Silverman, B., Ramsay, J., 2005. Functional Data Analysis. Springer. +Smale, S., Zhou, D.X., 2009. Geometry on probability spaces. Constructive +Approximation 30, 311–323. +Stoyan, D., Kendall, W., Mecke, J., 1988. Stochastic geometry and its applica- +tions. Bull. Amer. Math. Soc 19, 520–523. +Stoyan, D., Stoyan, H., 1994. Fractals, Random Shapes and Point fields. Meth- +ods of Geometrical Statistics. John Wiley & sons. +Suquet, C., 1986. Espaces autoreproduisants et mesures al´eatoires. Ph.D. thesis. +Lille 1. +Venables, W.N., Ripley, B.D., 2013. Modern applied statistics with S-PLUS. +Springer Science & Business Media. +20 + diff --git a/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/load_file.txt b/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1851eaa9ce790448dc892ad455356bdf16731d9c --- /dev/null +++ b/Y9AzT4oBgHgl3EQf2P5J/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf,len=889 +page_content='On using Reproducible Hilbert Spaces for the analysis of Replicated Spatial Point Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Sim´o Department of Mathematics-IMAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Universitat Jaume I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' del Riu Sec s/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 12071-Castell´o, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' January 6, 2023 Abstract This paper focuses on the use of the theory of Reproducing Kernel Hilbert Spaces in the statistical analysis of replicated point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We show that spatial point processes can be observed as random variables in a Reproducing Kernel Hilbert Space and, as a result, methodological and theoretical results for statistical analysis in these spaces can be applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In particular and by way of illustration, we show how we can use the proposed methodology to identify differences between several classes of replicate point patterns using the MBox and MANOVA tests, and to classify a new observation, using Discriminant Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' keyword Reproducing Kernel Hilbert Space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Functional Data Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Repli- cated spatial point processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Analysis of Variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Supervised Classifica- tion 1 Introduction A spatial point process is a stochastic random process whose realizations are locally finite sets of points (locations of events) in a study region E of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The term locally finite means that for any Borel bounded set there is a finite number of points (with probability one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Spatial point patterns arise as the natural sampling information in many problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Examples include the positions of trees in a forest, galaxies in the sky, a certain type of commerce in a city or cases of a certain disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Seminal books on the theory of point processes and their applications include Stoyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Stoyan and Stoyan (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Baddeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Illian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Diggle (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Cressie (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In mathematical terms, if (E)n ≡ E • E • · · · • E (n times) is the set of n elements of E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We define the exponential space as Ee ≡ ∞ � n=0 (E)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='01811v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='ME] 4 Jan 2023 A point process, Φ, is defined as a measurable application of a probability space (Ω, A, P′) on the measurable space (Ee, Be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For details on the measurable exponential space see Carter and Prenter (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Accurate and well-founded methodologies for the analysis of point processes are widely used in the literature, but most of them focus on the case where only one observation of the process is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' There are mainly two different approaches to represent and/or describe point processes: event densities and distributions and random counting measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In this paper we focus on the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Our goal is to take advantage of the relationship between random measures and RKHS-valued random variables, to work with point processes characterized by random functions in a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Thus, we explore how we can deal with more complex statistical methodologies in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In particular, our proposals will make it very natural and straightforward to analyze replicated point patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', data sets consisting of several point patterns that can be considered independent replicates of the same experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' RKHS spaces are well known in the statistical literature, and are frequently used in the context of Machine Learning (Berlinet and Thomas-Agnan, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Saitoh and Sawano, 2016) as a valuable tool in classification or regression prob- lems on Euclidean or L2 spaces or even on Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In particular, in Euclidean spaces the success of many classification algorithms is due to the use of kernel methods (Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' However, there is hardly any lit- erature on statistical methods when the data live in a RKHS (Luki´c and Beder, 2001), which is our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The theory of statistics with functional data is an important field of research in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' It deals with samples in which a function is observed for each indi- vidual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The books by Silverman and Ramsay (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Ferraty and Vieu (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Horv´ath and Kokoszka (2012) and Aneiros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (2017) are key references, as are the excellent reviews by Cuevas (2014) and Goia and Vieu (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Although the theory of functional data analysis has incorporated many tools from classical parametric or nonparametric statistics, the infinite-dimensional nature of the sample space poses particular problems (Ferraty and Vieu, 2006), even though, in practice, one has only sampled observed curves into a finite set of observation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' There are two different perspectives on functional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The first view is that functional data are realizations of random variables taking values in a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The second view is that functional data are the sample trajectories of a stochastic process with smooth mean and covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' There are subtle differences between the two perspectives from a theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' RKHSs are often present in the second perspective of functional data analysis (Preda, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Kadri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2016) since the Lo´eve-Parzen congruence (Aronszajn, 1950) links a second-order stochastic process with the RKHS generated by its covariance function (Eubank and Hsing, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Kupresanin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The functional principal component directions turn out to be an orthonormal ba- sis of the Hilbert-Schmidt covariance operator associated with the covariance 2 kernel (Horv´ath and Kokoszka, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Cuevas, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Although, in our case the functional data be by definition a sample of a variable in a Hilbert space (first point of view), our approximation will be similar to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Baddeley (2015), the practical analysis of replicated point patterns is ex- plained using the R package Spatstat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In particular, one of the dataset included in this package will be used in this paper: the Pyramidal dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' This dataset contains data from Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1991) and they are locations of pyramidal neurons in human brain of 12 normal, 9 schizoaffective, and 10 schizophrenic human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' All our implementations were written with R (R Core Team, 2021), mainly using the Spatstat and the MASS packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The article is organized as follows: First at all, Section 2 concerns the theoretical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Secondly, in Section 3 the theoretical concepts are applied to the statistical analysis of replicated point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' After taht, two particular applications are detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Fi- nally, conclusions are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 2 From point processes to random elements in a reproducing kernel Hilbert space As indicated in the introduction, the objective of this paper is to show a method- ology that allows the study of point processes by characterizing them by means of random functions in a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In this section we introduce the theoretical concepts necessary for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' First, the definition and properties of re- producible kernel Hilbert spaces are briefly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Second, we see how to embed measures in a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Thereafter, we express a point process as a random measure and discuss some theoretical results on random variables in Hilbert spaces, in general, and RKHS, in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='1 Reproducible kernel Hilbert spaces A Reproducible Kernel Hilbert Space (RKHS) is a Hilbert space of functions f : E → R with some practical and interesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The theory of Reproducible Kernel Hilbert Spaces was developed by Aronszajn (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Let H be a Hilbert space of real-valued functions defined on E and ⟨·, ·⟩H the inner product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A function k : E × Rn → R, is said to be an reproducing kernel (rk) associated with H if it satisfies: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' for every x ∈ E, k(·, x) ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' k satisfies the ”reproducing property”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' that is, ∀f ∈ H and x ∈ E f(x) = ⟨k(·, x), f⟩H Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A Hilbert space of real-valued functions is a Reproducible Kernel Hilbert Space if it has a reproducing kernel (rk) associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 3 A RKHS can be obtained from a kernel and each kernel determines (Moore- Aronszajn theorem) a unique RKHS, denoted by Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The construction of Hk is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We consider the set H0 of linear combinations: H0 = { N � i=1 bik(·, xi), N ≥ 1, bi ∈ R, xi ∈ E}, the RKHS H associated with the kernel k is the closure of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Given φ1 = �N1 i=1 aik(·, xi) and φ2 = �N2 i=1 bik(·, yi), ⟨φ1, φ2⟩k = � i � j aibjk(xi, yj) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2 Embedding measures in a RKHS The study of random measures requires sophisticated mathematical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For this reason, and following Berlinet and Thomas-Agnan (2011), we first show how reproducing kernels can be used to represent measures in functional spaces starting with Dirac measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We will then use this embedding to define inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Given a compact subset E of R2, and B the σ−algebra of Borel of subsets of E, the Dirac measure δx is defined for x in E by: δx(A) = � 1 if x ∈ A 0 if x /∈ A where A is a Borel set in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The mapping: δx �→ k(·, x) embeds the set of Dirac measures on E in the RKHS with kernel k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' If the function k(y, ·) is measurable, the value k(y, x) of the function k(·, x) at the point y can be written as the integral � k(y, t)dδx(t), and the mapping can be rewritten as: δx �→ Iδx = � k(y, t)dδx(t) In addition, for any measurable function f in Hk ⟨f, k(·, x)⟩H = f(x) = � f(t)dδx(t) More generally, if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', xN are N distinct points in E and b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', bN, N non null real numbers, a linear combination µ = N � i=1 biδxi of Dirac measures is called finite support signed measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We can extend the previous mapping with µ �→ N � i=1 bik(·, xi) = � k(y, t)dµ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1) 4 This mapping embeds in Hk the set of measures on E with finite support, M0, and the set H0 can be seen as the set of its representers in Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Again, we have the property that for any measurable function f in Hk ⟨f, �N i=1 bik(·, xi)⟩H = �N i=1 f(xi) = � fdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Following the generalisation, we can also embed the set M of signed measures on E in Hk with the mapping: M → Hk µ �→ Iµ = � k(·, t)dµ(t) see Berlinet and Thomas-Agnan (2011) to more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' If we assume that k is such that the functions Iµ and Iν are different if µ and ν are not equal, Theorem 99 of Berlinet and Thomas-Agnan (2011) guaranties that the mapping: M × M → R (µ, ν) �→ ⟨Iµ, Iν⟩H defines an inner product on M for which M0 is dense in M and its converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Guilbart (1979) was the pioneer in studying the relationships between repro- ducing kernels and inner products on the space M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' He exploited the embedding and characterized the inner products inducing the weak topology on sets of mea- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='3 Point processes and random measures If Φ is a point process, the random counting measure associated to Φ is defined as: µΦ(B) = #{x : x ∈ Φ ∩ B}, (2) for B a Borel set in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The random counting measure of a point process characterizes its probabil- ity distribution and provides a framework for developing the theory of point processes as part of a general theory of random measures Daley and Vere-Jones (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' If M is a set of measures in E equipped with some σ-algebra, a random measure can be regarded as a random variable with values in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For a detailed study of the theory of random measures, see Daley and Vere-Jones (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Although this concept seems easy, the very definition of random measures raises delicate problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For instance, the definition of the σ-algebra possibly derived from some topology on M is not a simple matter and the resulting theory involves delicate mathematical questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For this reason, once we have seen how we can embed the set of measures in a RKHS, we use this embedding to define and study random measures as random elements in a RKHS i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' RKHS-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We will assume that 5 the random variable takes its values in M (or M0) with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' This is the construction proposed by (Suquet, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' With this construction, we can subsequently exploit the known results con- cerning the probability laws in a separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In addition, RKHS are vectorial metric spaces and they can be considered as the natural extension of the usual Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Most of the theoretical and methodological sta- tistical results defined in Euclidean spaces are directly inhered in RKHS spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Furthermore, the completeness of Hilbert spaces gives a framework in which to work with infinite-dimensional vectors as the limit of finite-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Probability theory in Banach and Hilbert spaces is an important branch of modern probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A complete treatment of this topic can be found in Ledoux and Talagrand (1991) and Hsing and Eubank (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A large number of concerning large sample results can be applied on a separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Among all these theoretical results, we recall here only a central limit theorem which guarantees convergence to a Gaussian process analogous to Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' See Hsing and Eubank (2015) for a complete exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Let χ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', χn be independent and identically distributed random elements in a Hilbert space H with mean 0 and E∥χi∥2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Then ξn := �n i=1 χi � (n) →d ξ, where ξ is Gaussian random element of H with covariance operator equal to E(χi ⊗ χi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Being ⊗ the tensor product operator in H, ∥ · ∥ the norm defined by the interior product in H and →d denotes convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Furthermore, if the Hilbert space is also a RKHS, we have more important and strong properties which guarantees the application of standard statistical techniques to random variables in a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Guilbart (1979) proved a Glivenko- Cantelli theorem that he applied to estimation and hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Berlinet (1980b,a) studied weak convergence in the set of probabilities on a RKHS, mea- surability and integrability of RKHS-valued variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Another important re- sult, although it will not be used in this paper, is the theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='1 of (Hsing and Eubank, 2015), which tells us that a random variable in a RKHS is also a stochastic process and the reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Turning to the case of point processes, Equation 2 can be rewritten as: µΦ(B) = � xi∈Φ δxi(B) = N � i δxi(B), (3) for B a Borel set in E and where δ denotes the Dirac measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The counting measure defined from Φ is a random measure and all previously mentioned for random measures applies to the statistical analysis of replicated point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 6 A counting measure is a particular case of finite support measure and we can embed it in a RKHS using the mapping 1: µΦ �→ N � i=1 k(·, xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (4) And, we will assume that its associated RKHS random variable takes its values in the set of counting measures on E, N ⊂ M0, with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Moreover, it is satisfied that if two counting measures µ and ν are not equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', they have different supports {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', xN} and {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', yM}, Iµ = �N i=1 k(·, xi) and Iν = �M i=1 k(·, yi) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Regarding computational aspects, the mapping 4 can be easily obtained using, for example, the function density of the spatstat package of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Although the possibilities opened up by this way of working are enormous, in this paper, we just focus on two applications for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In the first application, we have different experimental groups, we observe independent replicates of a point process within each group and we are interested in contrasting whether there are differences between groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' an ANOVA problem where the response is a point pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In the second example, we also have different groups but we are interested in a rule to classify a new point pattern in one of these groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' a supervised classification problem when the explanatory variable is a point pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In both cases, classical multivariate statistical methods will be used in order to take profit of all the theoretical results aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 3 Application to the statistical analysis of repli- cated point processes In the previous section, we have seen how to transform point processes into random elements in a RKHS ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' therefore, in this section, we assume that we have a sample {ϕi(·)}n i=1 of a random element in the RKHS Hk, each of the elements of the sample having the expression: ϕi(x) = Ni � l=1 k(x, xil), x ∈ E (5) Because our data are a particular kind of functions, it would make sense to use the insights of functional data analysis (FDA) to carry out any statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' But, our data are very different to the typical data in FDA issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Firstly, our data are not expressed in the standard form, where the i-th func- tional datum is given just by a set of discrete measured values ((yi1, t1), · · · , (yimi, tmi)) with little knowledge of the analytical form of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Each datum of our sample is a function ϕi(x) whose analytical expression is defined by (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 7 Secondly, our function space is a RKHS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' our raw functional data “live” in a RKHS and, as it was said before, its vector structure and inner product can be exploited in the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For these reasons, our proposal is to express each function of our dataset with respect to the orthonormal base given by the eigenfunction decomposition of the kernel that defines the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The elements given by the eigenfunction decomposition of the kernel are orthonormal and are ordered according to an optimality approximation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' These properties allow us to reduce the dimension and thus we can apply classical statistical procedures as in the multivariate Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Similar ideas were previously used by authors in a very different context: supervising classification of geomet- rical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The following results are similar to results of Section 3 in Barahona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' To obtain the projection on this base, we need first to change the expression of our functions so that the points at which the kernel is evaluated are the same in all the point patterns of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For that we can use the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Theorem 4 (Representer Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Given ϕi(·), a grid {al}N l=1 in E, denoting ϕi(al) = bil and given a regularization parameter γ > 0 then ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='χi : R2 → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' χi(y) = �N l=1 βilk(y, al) such as: χi = arg min g∈Hk 1 N N � i=1 (g(al) − bil)2 + γ ∥g∥2 Hk , (6) where βil ∈ R (for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' , N) are the solutions of: (γ N IN×N + K|a)βi = bi, with K|a the matrix defined as (K|a)(i, j) = k(ai, aj), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' , N, and βi, bi are the N × 1 vectors βi = � � � � � βi1 βi2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' βiN � � � � � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' bi = � � � � � bi1 bi2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' biN � � � � � As a result of applying the theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' from now on we will work with the sample: {χi(·) = N � l=1 βilk(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' al)}n i=1 (7) It is well known (Hsing and Eubank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 2015) that if E is compact and k continuous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' measurable and bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' the integral operator associated to the kernel function k and defined by: (Kf)(·) = � E k(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' x)f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' f ∈ L2(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (8) 8 where L2(E) is the space of square integrable functions on E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' is a compact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' continuous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' self-adjoint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' and positive operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As a result, it can be expressed as K = � q≥1 λqeq ⊗ eq, where {λq, eq}q is the countable sequence of its eigenvalues and (orthonormal) eigenfunctions (spectral decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In addition: k(x, y) = � q≥1 λqeq(x)eq(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' And { � λqeq}q is a complete orthonormal basis for Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The following results show us how we can project our sample in this base of the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Let χi be an element of the RKHS Hk, it can be expressed as χi(·) = ∞ � q=1 µqi �� λqeq(·) � , (9) where { � λqeq}∞ q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In addition, if χi(·) = ∞ � q=1 µqi �� λqeq(·) � and χj(·) = ∞ � q=1 µqj �� λqeq(·) � , the inner product is: ⟨ϕi, ϕj⟩k = � q µqiµqj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Furthermore (Smale and Zhou, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Gonz´alez and Mu˜noz, 2010), the first d = rank(K|a) coefficients µqi can be approximated by � µqi = � ℓq(vq · βi) (10) where vq ∈ RN are the eigenvectors of K|a, ℓq are the eigenvalues of K|a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The following result assures us that if we express our infinite-dimensional data in this basis and truncate it to obtain a finite-dimensional vector sample, we will have the highest possible accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 in Hsing and Eubank (2015) tell us that for a fixed integer r > 0 with λr > 0: min f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=',fr∈Hk � � E×E � k(y, x) − r � q=1 fq(y)fq(x) �2 dydx, the minimum is achieved by fq = eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 9 This result ensures that the truncated eigenvalue-eigenvector decomposition provides the best approximation to k and, as a result, to our data (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' If we truncate the summation in Equation 9 to a low number of terms r, r ≤ d = rank(K|a), each point pattern ϕi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' , n, is given by the coefficients µqi for q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' , r (estimated by � µqi), on the orthonormal basis { � λqψq}∞ q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As a result, it can be represented as the r-dimensional vector µi = ( � µ1i, � µ2i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' , � µri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (11) This expression optimally reduces the infinite-dimensional problem to a finite-dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' It is now possible to apply well-known classical multivariate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In particular, in the following section two well-known classical multivariate methods will be used for illustrative purposes: MANOVA and Discriminant Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 4 Application to locations of pyramidal neurons in human brain As mentioned before, the Pyramidal dataset is included in the R package spastat and contains data from Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The data are the locations of pyramidal neurons in human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' One point pattern was observed in each of 31 human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' There were 12 normal control, 9 schizoaffective, and 10 schizophrenic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' See Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1991) for a more detailed explanation of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Figure 1 gives plots of the point patterns in our sample, one plot for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In this example E = [0, 1] × [0, 1] and a gaussian kernel with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='05 were used to embed the point patterns in a RKHS using the map 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A grid {al}N l=1 with al equally spaced in E with a vertical and horizontal step of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='02 is used to apply the representer theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The parameter γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='000127 in Equation 6 is fixed to the minimum value wich makes matrix (γ N IN×N + K|a) invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The parameters h and σ are related with the accuracy of our working data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' When h and σ decrease, we have more accuracy, but more computational cost and complications in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The chosen values represent a balance between both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' After applying the representer theorem we have a sample {χi(·), yi}31 i=1 with χi(·) = �N l=1 βi,lk(·, al), and yi a categorical variable indicating if the subject is normal, schizoaffective or schizophrenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Figure 2 we can see the plot of one point pattern of each group and its corresponding elements ϕi and χi in the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Finally, Equation 9 is applied to obtain the vector of coefficients µi that represent each point pattern in relation to the orthonormal base of Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Because the size of our sample is very small (12, 9 and 10 cases in each group), we truncate it at r = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' It is worth analysing the meaning of the firsts functions of the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Figure 3, we can see the six firsts eigen functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The coefficient of the first 10 Figure 1: Positions of the pyramidal neurons for the three groups of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 11 Normal 1 2 3 4 o o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 00 o0 0 00 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 0 0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 o 8 0 18 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='o 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 20 00 Pe 8 0 5 6 7 8 o 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 00 0o 0 o0 0 c00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=',0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='80 00 0 00 0 T8 o° 88 0 0 0 6 10 11 12 0 0o 0 o0 0 0 0 00 0 0%0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='°。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 0 0 00 0 ° 08080 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='88 608 80 0 0 oo 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=': 0 0 0Schizoaffective 13 14 15 9 0 0 0 0 0 00 0 o0 0 0 16 17 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 0 0 00 o 0 00 18 960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='000 30 19 20 21 0 0 o 0 00 og 0 0 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 00 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 0 0 0 oo 00%0 0 8 08 eSchizophrenic 22 23 24 25 68 +00 0 0 0 00 0 %00 0 0 000 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 90 8 26 27 28 29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='% 004 0o9 800 000 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 8 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='%0 00 0 0 0 80 0 00 00 8 0 0 30 31 00 0 % 0 o% 0function measure high density concentrated in the center of the window, the second and third high density just within one half of the window and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='1 ANOVA Two principal approaches can be found in the literature of spatial point pat- terns to identify significant differences between several experimental groups (Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The first one (Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Ram´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Gonz´alez Monsalve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2018) is based on functional descriptors of the pat- tern and nonparametric inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The second one is based on assuming a parametric model for the pattern (usually pairwise interaction point process or Gibbs processes), the parame- ter of the models are estimated using maximum likelihood or pseudo-likelihood methods for each group and differences between groups are tested by compar- ing fits with and without the assumption of common parameter (Illian and Hendrichsen, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Illian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Following the aforementioned procedure, we will work with the RKHS ele- ments resulting of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Since a RKHS is a vector space, the mean sample element is simply obtained as: ¯χ(·) = �N l=1 �n i βi,l n k(·, al).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Figure 4 we can see the mean sample element of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In this example, we are only focused on the answer to the following question: do the observed patterns differ significantly in mean from group to group?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For this purpose, a Multivariate ANOVA test can be applied to the r-variate sample {µi, yi}31 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Before the MANOVA test, a Box’s M test must be applied to test the assumption of homogeneity of variance-covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The results of both tests can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' At a significance level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='05, the hypothesis of homogeneity of variance-covariance matrices would be accepted and the hypothesis of equality of means would be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' It is important to note that, in this case, the sample size is small and there- fore, the Central Limit Theorem is not applicable to assume normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Shapiro univariate tests applied to the residuals gave non significative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Although these results do not guarantee multivariate normality, is not a cause for con- cern for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The first reason is the robustness to non-normality of the MANOVA test, and the second is that this example is only illustrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Once the multivariate hypothesis of equality of means has been rejected, we perform an univariate ANOVA test to analyse with more detail the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The respective p-values for the first six coefficients of our base were: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='069, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='066, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='42, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='115, and we can conclude that the differences between classes are mainly in the four coefficients corresponding to the four first functions of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' It is important to emphasize again that, our objective in this section is to show the potential applications of the proposed methodology, a deeper study conducted in conjunction with experts in neuroanatomy would be necessary to reach clinical conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 12 Figure 2: Point pattern and RKHS functions of the first subject of each class before and after applying the representer theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Test Statistic df p-value Box’s M-test 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='009 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='7304 Manova 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2358 12-48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='02451 Table 1: Box’s M and MANOVA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 13 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0Figure 4: The means of the RKHS functions of each class: normal, schizoaffec- tive and schizophrenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2 Discriminant Analysis Once significant differences between groups of subjects have been found, it could be very useful to find a decision rule to classify a new individual as normal, schizoaffective or schizophrenic on the basis of its spatial pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' To our knowledge, the literature about supervised classification methods applied to replicated point patterns is scarce and relies mainly on dissimilarity- based methods (Mateu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Pawlasov´a and Dvoˇr´ak, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Similar methods have been used for supervised classification of germ-grain models in Gallego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As it is well known, the literature on supervised classification methods is extensive and covers a large number of methods ranging, from those based on multivariate statistics to the most modern deep learning techniques (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' But, as mentioned before, in this paper, we focus only on classical multivariate statistical methods, since the theorem 3 would allow us to take ad- vantage of probability parametric models as well as convergence theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' For this reason, linear and quadratic discriminant functions (Venables and Ripley, 2013) will be used to illustrate the new methodology proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As it is known, both are parametric Bayesian methods that assume a multivari- ate Gaussian model for the explanatory variables, with and without equality of variance-covariance matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The a priori probabilities of each class will be estimated from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Since our dataset is very limited, we will first present some illustrations using simulated point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Two different experiments were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In the first experiment two samples of size 20 of an homogeneous Poisson point process (HPPP) with different intensities, λ1 and λ2, were simulated in a rectangular window of size one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' This experiment was repeated two times with λ1 = 50 and λ1 = 90, respectively and λ2 = 100 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Figure 5, we can see one point pattern of each class and its corresponding element in the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Since the difference between the two classes of point processes is in the 15 00 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content="08'0 6 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0mean of its counting measures, linear discriminant functions have been used in both cases (lda function of R package MASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The Table 2 shows the training errors and the cross-validation errors, excellent results have been obtained, even in the second case in which the difference between both models is very slight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In the second experiment, two samples of size 30 of two different point pro- cesses were simulated again in a [0, 1] × [0, 1] window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The first sample cor- responds to a homogeneous Poisson point process with intensity λ1 = 36 and the second to a Poisson cluster point process (PCPP) with the intensity of the Poisson process of centres κ = 6, and each cluster consisting of 6 points in a disc of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The resulting intensity is also λ2 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In Figure 5, we can see again one point pattern of each class and its corresponding element in the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In this case, both point processes have the same intensity and the differ- ence between both classes is given by the spatial variability, for this reason the linear discriminant function does not work well and a quadratic discriminant function must be used (qda function of R package MASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Figure 5: One simulated point patterns and its corresponding RKHS element of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' First row: homogeneous Poisson point processes with λ1 = 50 and λ2 = 100, second row: homogeneous Poisson point processes with λ1 = 90 and λ2 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Third row: homogeneous Poisson point process with λ1 = 36 and Poisson cluster point process with λ2 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 16 0 00 0 80 0 0 00 00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 00 CPO 8 0 % 8°00 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content="4 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 0 2 0 0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='2 04 06 08 10The same accuracy parameters of the previous section have been used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='02 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Regarding to the number of variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' the value of r, different values ranging from r=6 to r=10 have been tested, but no major differences have been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In general, the best results were obtained for r=7, and these are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The results are again excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In addition to obtaining very small cross-validation errors, the a posteriori probabilities of well-classified cases are all practically greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='95 and those of misclassified cases lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Models Intensisties Training error CV error HPPP-HPPP λ1 = 50, λ2 = 100 0 0 HPPP-HPPP λ1 = 90, λ2 = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='1755 HPPP-PCPP λ1 = 36, λ2 = 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='117 Table 2: Supervised classification errors in the simulated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' After testing the performance of the proposed methodology on a super- vised classification problem, the lda function was applied to our real example to find a decision rule to classify a new individual as normal, schizoaffective or schizophrenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We used linear discriminant analysis because the Box’s M test, used to test the hypothesis of homogeneity of variances in the previous section, did not yield significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As expected, no good results were obtained due to the limitations of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' The training error was relatively small (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='29), but the cv error was not at all satisfactory (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' A larger dataset would be necessary in order to be able to use in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' In our view, our methodology could be used without modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 5 Conclusions We have introduced a new methodology for the statistical analysis of replicated spatial point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' This methodology is based on the fact that the proba- bility distribution of a point process is completely determined by its associated random counting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Random measures can be embedded in a RKHS and, in this way, we transform the point process in a random element in a RKHS, where theoretical founded methods and algorithms can be applied, similar to what is done in an Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' To do so, we express our data in the base given by the kernel’s eigenfunctions and truncate this expression in the required dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' This guarantees to move to a lower dimension with the least loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' As an example of the potential real-life applications of the proposed method- ology, we have used it to detect differences between point patterns of pyramidal neuron locations in the human brain from three groups of subjects (Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' We have also used it to classify new observations using several simu- lated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' With the results of these experiments, it can be stated that our methodology is feasible for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 17 References Aneiros, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', Bongiorno, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', Cao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', Vieu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2017.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Modern applied statistics with S-PLUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQf2P5J/content/2301.01811v1.pdf'} diff --git a/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf b/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d9b322c3788256f0f0bd5e386aa11b824962c2f8 --- /dev/null +++ b/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf 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Degree Two +Henning Fernau (Universit¨at Trier, Universit¨atsring 15, Trier, Germany)1, +Kshitij Gajjar (Indian Institute of Technology Jodhpur, Rajasthan, India)1 +Abstract +The concept of sum labelling was introduced in 1990 by Harary. A graph +is a sum graph if its vertices can be labelled by distinct positive integers in +such a way that two vertices are connected by an edge if and only if the +sum of their labels is the label of another vertex in the graph. It is easy to +see that every sum graph has at least one isolated vertex, and every graph +can be made a sum graph by adding at most n2 isolated vertices to it. The +minimum number of isolated vertices that need to be added to a graph to +make it a sum graph is called the sum number of the graph. +The sum number of several prominent graph classes (e.g., cycles, trees, +complete graphs) is already well known. We examine the effect of taking +the disjoint union of graphs on the sum number. In particular, we provide a +complete characterization of the sum number of graphs of maximum degree +two, since every such graph is the disjoint union of paths and cycles. +Keywords: +Sum labelling, Sum number, Cycles, Paths, Graph union +1. Introduction +The area of graph labelling is a specific subarea of graph theory that has +developed an enormous body of literature, as testified by Gallian’s dynamic +survey [1] which mentions over 3000 research papers. One of these labellings +is sum labelling, introduced by Harary [2] as a form of representing graphs. +It is known [3] that every n-vertex graph G can be represented via a sum +labelling, which means that it is possible to add at most n2 isolated vertices +(also called isolates, in short) to G to make it a sum graph. This makes sum +Email addresses: fernau@uni-trier.de (Henning Fernau (Universit¨at Trier, +Universit¨atsring 15, Trier, Germany)), kshitij@iitj.ac.in (Kshitij Gajjar (Indian +Institute of Technology Jodhpur, Rajasthan, India)) +Preprint submitted to Discrete Mathematics +January 6, 2023 + +1 +3 +2 +5 +4 +1 +4 +3 +7 +5 +(a) +(b) +(c) +Figure 1: (a) This graph is not a sum graph, because it has no isolated vertices; (b) This +is an incorrect sum labelling of a sum graph, because (1, 4) is not an edge yet there is a +vertex labelled 1 + 4 = 5 in the graph; (c) This is a correct sum labelling of a sum graph. +labelling a compelling concept from the viewpoint of computer science also, +because it may be that certain graphs can be encoded much more succinctly +with sum labellings than with the more traditional ways of storing graphs. +Let us now fix some notations. We deal with simple, undirected graphs, +specified (as usual) as G = (V, E), where V is the (finite) set of vertices of +G, and E is its set of edges. If v is an endpoint of an edge e, then we say +that v and e are incident. The number of edges incident to a vertex is the +degree of the vertex. Let N denote the set of all natural numbers (positive +integers). Then, we say that G is a sum graph if there exists an injective +mapping λ : V → N (called the sum labelling of the vertices of G) such that +E = {xy | ∃z ∈ V : λ(z) = λ(x) + λ(y)}. +Up to isomorphism, the set of numbers λ(V ) therefore determines G. In +other words, λ encodes G. As isolated vertices (i.e., vertices of degree zero) +are usually irrelevant in applications, λ(V ) can be viewed as the description +of G\I, where I is the set of all isolated vertices of G. Then, λ(V ) is called +the sum number encoding of G \ I. Conversely, given a graph G without +isolates, the minimum number of isolates that need to be added to G in +order to make it a sum graph is called the sum number of G, written as +σ(G). Thus, G + Nσ(G) is a sum graph. (Here, + denotes the disjoint union +of graphs. Also, Ni denotes the null graph (edgeless graph) on i vertices, +or equivalently, a set of i isolated vertices.) See Figure 1 for some examples +and non-examples of sum graphs and sum labellings. +A labelling function λ can be also seen as operating on edges by the +summability condition. λ(e) for an edge e = xy ∈ E is defined as λ(x)+λ(y). +Thus, though only the vertices are labelled by a sum-labelling, we sometimes +2 + +also refer to its edges as labelled by the sum of its endpoints (two different +edges can have the same edge label). +Are substantial savings possible with sum number encodings of graphs? +Some partial answers are possible from the literature. For instance, σ(Kn) = +2n−3 is known for n ≥ 4, i.e., 3n−3 numbers suffice to store the information +about the complete graph Kn, while traditional methods would need O(n2) +bits. As mentioned in [4], this can be obtained by labelling vertex xi with +4i − 3, with 1 ≤ i ≤ n, leading to isolate labels 4j + 2 for 1 ≤ j ≤ 2n − 3. +Hence, the sizes of the labels are in fact linear in n. +The focus of our study is the sum number of certain graphs. This follows +much of the tradition in the literature, as can be seen in surveys like [1, 5]. +More precisely, we prove as our main result a complete picture of the sum +number of every graph of maximum degree two. As a consequence, if G has +maximum degree two, then σ(G) ≤ 3. This is not completely expected, as +it is known that the sum number of general graphs grows with the number +of edges [6]. In fact, this can happen even with sparse graphs [7, 8]. +When talking about sum labelling a whole infinite family of graphs G, +often with the additional property that for each positive integer n, there is at +most one graph Gn of order n within G, we also speak of a labelling scheme +λ : N → N that formalizes the labelling strategy that we suggest for G in the +following sense. For Gn, to be labelled with i isolates, we take {1, . . . , n} as +the vertex set of Gn and consider the set of numbers {λ(1), . . . , λ(n+ i)} as +the set of labels of the sum graph Gn +Ni. Extending this notion, a general +labelling scheme is specified by three functions λ : N → N, σ : N → N and +ι : N → N that are interpreted as a labelling strategy for Gn ∈ G of order n, +with i isolates, as follows. As the set of vertices of Gn + Ni, we consider +Vn+i = {σ(n), σ(n) + 1, . . . , σ(n) + n − 1, ι(n), ι(n) + 1, . . . , ι(n) + i − 1}, +where the first n numbers denote the vertices of Gn, and as labels we take +λ(j) with j ∈ Vn+i. This boils down to a labelling scheme if σ(n) is constant +one and ι(n) = n + 1. More general labelling strategies of n-vertex graphs +of a family of graphs G are possible and will be discussed later in this paper. +Our main result is a complete precise characterization of all graphs G +of maximum degree two: +Theorem 1. Let G be a graph of maximum degree two. Then, σ(G) = δ(G) +except for two graphs, namely C4 and C4 + P2, for which σ(G) = δ(G) + 1. +Harary [2] already showed that σ(C4) = 3, and that the minimum degree +of a graph is always a lower bound on its sum number (i.e., σ(G) ≥ δ(G) +3 + +Sum-labelling graphs of maximum degree two: a strategy +1. Firstly, we deal with all cycles of length not equal to four (if any), +in descending order of length. +2. Secondly, we deal with all cycles of length four (if any). +3. Finally, we deal with paths (if any), in descending order of length. +Figure 2: Our proposed strategy for sum-labelling graphs G with 1 ≤ δ(G) ≤ ∆(G) ≤ 2. +for all graphs G). Therefore, to prove our main theorem, it suffices to show +that σ(G) ≤ δ(G) for all graphs G of maximum degree two, except for C4 +and C4 + P2. An additional proof is required to show that σ(C4 + P2) = 2. +Apart from having a combinatorial result, we can also interpret our proof +as providing an algorithm that labels any graph of maximum degree two +optimally with respect to its sum number. +For the motivation of efficiently storing graphs, this is not completely +satisfying, as the sizes of the labels could be exponential in the number +of vertices of the graph according to our constructions, which means that +we might need up to O(n2) many bits for storing an n-vertex graph. In +principle and in general, we can do this more efficiently in terms of label +sizes [9], but the algorithm presented in [9] is not tailored towards using as +few isolates as possible, i.e., it does not obey the sum number of the graph, +which is the focus of this study. +Notice that every graph of maximum degree two is a disjoint union of +cycles and paths (in other words, each connected component of the graph is +either a path or a cycle). To prove our main theorem, we will deal with the +connected components in a specific sequence. This naturally produces an +algorithm that optimally labels (with respect to the sum number) all graphs +with maximum degree two. We provide a sketch of our strategy in Figure 2. +Figure 2 also explains the sequence in which we will treat all graphs of +maximum degree two. For example, if the graph G is +G = 5C3 + 2C4 + C6 + 2C7 + 3C9 + 4P2 + P5 + 2P8 + P9, +then we will deal with the components of G in the following order: +3C9, 2C7, C6, 5C3, 2C4, P9, 2P8, P5, 4P2. +4 + +2. The space complexity of sum labelling +One of our motivations to return to sum labellings was the idea that one +can use them to efficiently store graphs. This idea was already expressed +in [3]. There, they consider the notion of the range r(λ) of a labelling λ, +which is defined as the difference between max λ(V ) and min λ(V ),1 with +r(λ) = max +v,v′∈V λ(v) − λ(v′) . +To clearly distinguish our notion of range from the ones mentioned in +footnote 1, let us introduce the sum range number rσ(G) of a graph G +as the smallest range of a labelling of a sum graph G + Nk for some k. As +eventually the range grows with the number of vertices, here we propose +two different ways of ensuring that the numbers involved do not grow too +fast. +To better motivate the introduction of these new graph parameters, let +us first analyze the sizes needed to store graphs in a database using a sum +labelling encoding. +A graph G = (V, E) on n vertices can be stored as +follows: We need O(log n) bits to store n itself, plus O(log log(max λ(V ))) +bits to store log2(max λ(V )), O(log σ(G)) bits to store the number of isolates +and then log2(2 max λ(V )) · (n + σ(G)) more bits for the (at best ordered) +list of numbers (vertex and isolate labels). In the end, we have to store a +list of n + σ(G) many integers, each with log2(2k) many bits, because edge +labels (e.g., labels of isolates) have value of at most 2k. +Instead, one could also first store the smallest label and then one would +only need log2(r) bits per number, where r is the range of the labelling. More +precisely, if we want to given an estimate of the number of bits needed to +store graph G = (V, E) with the labelling λ, we get the following formula. +2(log2 n + min +v∈V log2(λ(v))) + |λ(V ∪ E)| · log2(r(λ)) +(1) +Notice that although it looks beneficial to minimize |λ(V ∪ E)| by choosing +a labelling λσ that achieves σ(G), i.e., where |λσ(V ∪ E)| = |V | + σ(G), +there could be another labelling λ with |λ(V ∪ E)| > |V | + σ(G), but r(λ) +could be much smaller than r(λσ), potentially out-weighing the disadvan- +tage of needing more isolates. This is true in particular when r takes values +exponential in n, as for the Ellingham-labelling for trees [11]. +1 In [3] and also in [10], under the name spum, the mentioned difference is considered +only for labellings that attain the sum number. +5 + +Further stretching our notation, we will also consider rλ for a labelling +strategy λ, i.e., for a way to label a whole family of sparse graphs as de- +scribed above, so that rλ can be viewed as a mapping that associates to n the +largest range of any labelling of an n-vertex graph according to this strategy. +Hence, we can analyze the growth of rλ for certain labelling strategies. +What is the main purpose of a graph database? Clearly, one has to +access the graphs. A basic operation would be to answer the query if there +is an edge between two vertices. Now, if max λ(V ) is polynomial in n = +|V |, we can answer this query in time O(log(n)). Namely, assuming the +polynomial bound on the size of the labels, we would need time O(log(n)) +to add the two labels of the vertices, and we also need time O(log(n)) +to search for the sum in the ordered list of numbers, using binary search. +Otherwise, the additional time O(log(max λ(V ))) would be quite expensive, +probably making the idea of storing large graphs as sum graphs in databases +unattractive. Therefore, also the range of labellings should be considered. +Other parameters that measure the space consumption of storing graphs +even more accurately have been discussed in [9]. However, for the discus- +sions in this paper, the two parameters λ and r(λ) suffice, also because these +are more accessible from the combinatorial viewpoint that we consider here. +The main difficulty in dealing with the combinatorics of sum labelling +prevails also for these modified definitions, which is the question of how to +prove lower bounds. The only general assertion that is available is to say +that the sum number of a graph is at least as big as its minimum degree. +There are also generalizations of this observation based on degree sequences +(see [12, 4]), but this is irrelevant to us, as we consider graphs of bounded +degree. For instance, this means that the sum number of a collection of +cycles is at least two. But, as we see in the following, even proving that +certain collections of cycles have a sum number of two is far from trivial. +There are no really systematic tools available. +Regarding the notion of sum range number, it is nice to observe that the +proof of Theorem 2.1 of [10] concerning the spum of a graph is also valid in +our case (which is, as discussed above, a definitorial variation of spum), so +that we can state without proof the following result. +Proposition 1. Let G be a graph of order n with minimum degree δ(G) +and maximum degree ∆(G). Then, rσ(G) ≥ 2n − (∆(G) − δ(G)) − 2. +Observe that for regular graphs, the lower bound stated in the previous +proposition simplifies to 2n − 2. Unfortunately, even for our simple graph +families, we reach this bound only occasionally. +6 + +16 +31 +17 +30 +18 +29 +19 +28 +20 +27 +21 +26 +22 +25 +23 +24 +47 +(b) +(a) +2 +3 +5 +6 +11 +12 +23 +24 +47 +48 +95 +96 +191 +192 +383 +384 +767 +Figure 3: (a) The exponential labelling scheme; (b) The linear labelling scheme. +3. A first example: labelling a disjoint collection of edges +This section should be treated as an introductory example into the intri- +cacies of sum labelling. It has also been studied earlier [9, 10]. Moreover, it +covers an important subcase of our main theorem, which is 1-regular graphs, +or graphs of (maximum) degree one (without isolates). Also, one can see +examples that deal with the union of two graphs, each of sum number one. +It is known that all trees have sum number 1; according to a remark +following Theorem 5.1 in [11], all forests also have sum number 1. However, +it is not that clear how fast the label sizes grow in these constructions. Also, +recall that it is still an open question for general graphs with sum number +one whether their graph union again has sum number one [3]. Thus, we will +present two different constructions that label a disjoint collection of edges. +More mathematically speaking, we will show two labelling schemes for the +family of 1-regular graphs: an exponential labelling and a linear labelling. +3.1. An exponential solution +If you have n vertices (i.e., n/2 edges), label the first edge as (2, 3). The +second edge starts with the edge label of the first edge (2 + 3 = 5, so the +second edge is labelled (5, 6)). The third edge starts with the edge label of +the second edge (5 + 6 = 11, so the third edge is labelled (11, 12)), and so +on (see Figure 3 (a) for an example with n = 16). +Generalising this, the following labelling scheme λ : N → N works for +7 + +every 1-regular graph: +λ(n) = + + + +2 +if n = 1 +λ(n − 1) + 1 +if n is even +λ(n − 2) + λ(n − 1) +if n is odd and n > 1 +The Online Encyclopedia of Integer Sequences suggests that this is an- +other variation on Ulam numbers if we think of the starting point to be +λ(0) = 1. Then, λ(n) (for n > 1) can be seen as the smallest (when n +is even) or largest (when n is odd) number bigger than λ(n − 1) that is a +unique sum of two distinct earlier terms of the sequence. This connection +also suggests the following closed form: +λ(n) = +� 3 · 2k−1 +if n is even, i.e., n = 2k +3 · 2k − 1 +if n is odd, i.e., n = 2k + 1 +In other words, we have λ(n) ∈ Θ +��√ +2 +�n� +, implying that it is exponential +in n. Although the suggested labelling λ is optimal with respect to the sum +number σ, we see: r(λn) ∈ Θ +��√ +2 +�n� +. Can we do better with respect to +the sum range number? +3.2. A linear solution +Consider the following general labelling scheme for 1-regular graphs (ob- +serve that n is necessarily even) that we first describe in a more intuitive +fashion, already indicating the edges. +(n, 2n − 1), (n + 1, 2n − 2), . . . , +�3n +2 − 1, 3n +2 +� +. +Here, writing (λ(u), λ(v)) refers to two vertices u, v that are connected by +an edge (see Figure 3 (b) for an example with n = 16). Notice that all edge +labels sum to 3n − 1 (which is the isolate), and even the sum of the two +smallest labels, i.e., n + (n + 1) = 2n + 1, is smaller than 3n − 1 but bigger +than any other label in the graph. More formally, we consider the functions +λ, σ, ι with λ(n) = σ(n) = n and ι(n) = 3n − 1. This gives as vertex names +{n, n + 1, . . . , 2n − 1} for a 1-regular graph of order n. +This general labelling scheme can be further generalized by using the +parameters (x, y, d, k), with x < y (in our example, x = n, y = 2n − 1, d = +1, k = n/2 − 1), by putting +(x, y), (x + d, y − d), . . . , (x + kd, y − kd) . +8 + +All labels sum up to x + y, which is the isolate. As long as the sum of the +two smallest labels, i.e., 2x+d, is smaller than x+y but bigger than y, such +a sum labelling is valid. As the scheme consists of interleaving an increasing +arithmetic progression with a decreasing arithmetic progression (with the +same “slope”), we call such schemes arithmetic progression schemes. +The concrete arithmetic progression scheme that we first suggested has +as its range the numbers n through 2n−1 and is hence (nearly) optimal, as +Proposition 1 gives 2n − 2 as a lower bound. Singla, Tiwari & Tripathi [10] +show an upper bound of 2n−1. Therefore, we know that the optimal answer +is either 2n − 2 or 2n − 1, but we do not know which one it is. +3.3. Labelling paths +The ideas presented for 1-regular graphs work for paths also. As we +will need the exponential labelling scheme explicitly in the following, we +are going to present (only) this one now. For the linear solutions, we refer +to [9, 10]. +A scheme could be based on fixing two positive integers x, y as parame- +ters, and then defining the labelling scheme λφ +x,y : N → N as follows. +λφ +x,y(n) = + + + +x +if n = 1 +y +if n = 2 +λφ +x,y(n − 2) + λφ +x,y(n − 1) +if n > 2 +(2) +Due to the similarity to Fibonacci numbers, it is clear that λφ +x,y(n) = O(φn), +where φ is the golden ratio number, irrespectively of the start values x, y. We +can hence deduce the following well-known fact by this Fibonacci scheme. +Lemma 1. For any n ∈ N, σ(Pn) = 1. +4. Several labelling strategies for collections of cycles +Recall that according to the algorithmic strategy sketched in Figure 2, +we first deal with all cycles of length five and larger, then with all triangles, +and finally with all cycles of length four. The collection of C4 is the most +tricky one, as it could possibly leave us with three intermediate isolates. +Apart from this special situation, we will always face the situation that +after having dealt with k − 1 cycles, we have two isolates that we integrate +into the kth cycle as the start of a new Fibonacci-type labelling. This is +discussed in detail in the following subsections. +9 + +For the inductive argument, it becomes crucial to know that our la- +belling contains a non-trivial arithmetic progression, or NTAP for short. +This means that we find three labels x, x + d, x + 2d in the proposed la- +belling such that the offset d is not a label. +4.1. Collections of 4-cycles +In this subsection, we actually present two labelling strategies. +The +first one could be called “linear-exponential” in the sense that the proposed +labelling strategy is linear (an arithmetic progression) per cycle, but from +cycle to cycle, we observe an exponential growth. +It uses three isolates +(always) but has a smaller range compared to the second strategy that uses +two isolates only (from two C4 onwards) but needs a larger range. +4.1.1. A linear-exponential labelling scheme +Consider the labelling (2, 5, 8, 11) of a C4. Notice that the progression is +arithmetic, with a difference of 3. All numbers are congruent 2 modulo 3. +The three isolates are: (7, 13, 19). This arithmetic progression, with a +difference of 6, can be again lifted to a labelling of a second C4, which is +then (7, 13, 19, 25). All numbers are congruent 1 modulo 3. +The three isolates are now: (20, 32, 44). This arithmetic progression, +with a difference of 12, can be again lifted to a labelling of a third C4, +which is then (20, 32, 44, 56). All numbers are congruent 2 modulo 3, as +with the first C4. +It is clear that we can continue this construction by adding a fourth C4 +with labels (52, 76, 100, 124). All numbers are congruent 1 modulo 3. +To wrap up, the odd-numbered cycles get numbers that are congruent +to 1 modulo 3, while the even-numbered cycles get numbers which are con- +gruent to 2 modulo 3. These modulo 3 observations show that no edges can +ever occur between vertices in subsequent cycles. As all the edge labels of +the ith cycle can be found on the (i + 1)th cycle, we can see that (as the +differences on the ith cycle are of the form 3 · 2i−1), the non-edges (diag- +onals) on the ith cycle cannot be represented by vertices on the (i + 1)th +cycle. By the aforementioned exponential growth of the labels one cycle to +the next cycle, further non-edges cannot be represented by the suggested +numbers. This proves: +Lemma 2. If G is a disjoint union of C4’s, then σ(G) ≤ 3. +Moreover, we can state: +10 + +Lemma 3. rλ(G) ∈ O(2n/4) for a graph G of order n that is a union +of C4’s, for the specific labelling scheme λ that we described above. +4.1.2. Towards optimal sum labellings +We know that σ(kC4) ∈ {2, 3} (Lemma 2), and it is known that σ(C4) = +3. Can we possibly also show that σ(2C4) = 2 or even σ(3C4) = 2? Let us +try a bit of algebra, assuming arithmetic progression labellings of the two +considered C4’s. +x +x + d +x + 3d +x + 2d +2x + d +2x + 3d +2x − d +2x + 5d +4x + 4d +4x + 8d +Figure 4: An algebraic approach to the C4 problem. +The idea of Figure 4 is to find one of the three isolates of the second +cycle within the labels of the first cycle. The only way this could happen is +for the isolate 4x = (2x+d)+(2x−d). Clearly, 4x ̸= x. If 4x = x+d, then +3x = d. This contradicts the label 2x − d, which implies that 2x > d. If +4x = x + 2d, we conclude 3x = 2d, so that x is even and d is divisible by 3. +The smallest numbers satisfying these conditions are x = 2 and d = 3; see +Figure 5. These divisibility conditions also enforce that all other labellings +of this form have to be scalings of this minimal labelling by some constant +factor. Finally, if 4x = x+3d, then x = d. Hence, the number 2x+d = x+2d +would occur twice as a vertex label. Therefore, under the conditions that +our first cycle is labeled as in Figure 4, Figure 5 basically shows the only +possibility. Notice that this labelling contains the NTAP 2 − 5 − 8. +2 +5 +11 +8 +19 +13 +1 +7 +20 +32 +Figure 5: A minimal way to label a 2C4 with two isolates. +Could scaling help to also label 3C4 with our strategy? The somewhat +surprising answer is yes. First, we look at a concrete example in Figure 6. +The trick consists in the following steps: +11 + +1. Multiply all labels used so far by a sufficiently large constant z > 2, +which is four in our example. +We actually need that (modulo z) +z − 1 ̸= z + 1. To ease our inductive argument, let us always pick +z = 4. +2. Pick the smallest three labels of the first cycle, which is x = 8, x+d = +20, x + 2d = 32 in our example, and select numbers a, b, c, e to label +the third cycle. To avoid unwanted edges, choose a = x/2 + 1 (recall +that x must be an even number), b = x/2 − 1, c = (x + 2d)/2 + 1, +e = (x + 2d)/2 − 1. +3. Observe that the isolates of the 2C4-construction remain untouched. +4. Also, since our labelling of 2C4 contains a NTAP, the proposed la- +belling of 3C4 contains a NTAP too. +8 +20 +44 +32 +28 +52 +4 +76 +5 +3 +15 +17 +80 +128 +Figure 6: A way to label a 3C4 with two isolates. +32 +80 +176 +128 +112 +208 +16 +304 +20 +12 +60 +68 +17 +15 +63 +65 +320 +512 +Figure 7: A way to label a 4C4 with two isolates. +As the 2C4-construction remains untouched up to scaling, we can ac- +tually repeat this argument, which could give the labelling of a 4C4 as in +Figure 7, and this type of argument continues to prove by induction on k: +Lemma 4. σ(kC4) = 2 for all k ≥ 2. Moreover, the corresponding labelling +contains a NTAP. +Proof. Let us describe some details of the induction. For our inductive +argument to work, we make the additional claim that the three smallest +12 + +numbers x, y, z labelling the first cycle form an arithmetic progression, i.e., +there is a number d such that y = x + d and z = x + 2d. Moreover, d is not +a label of any vertex, so that the labelling satisfies NTAP. The induction +basis for k = 2 was given above and satisfies NTAP. Let us assume that the +labelling strategy works for some specific k = K ≥ 2. When we multiply all +labels of the first K cycles by four, then this will not change the fact that +(exactly) the edges of the K cycles are described by these numbers, plus +the two isolates that remain as isolates in the overall labelling. Also some +NTAP is found after the modification by multiplication. The labelling of +the (K + 1)st cycle builds upon the smallest three labels x, x + d, x + 2d of +the first cycle, choosing a = x/2 + 1, b = x/2 − 1, as well as c = a + d and +e = b + d as labels of the last cycle. As we multiplied all original numbers +by four, x is an even number. Also, a + b = x, a + e = b + c = x + d and +c + e = x + 2d, so that all wanted edge labels can be found as vertex labels +on the first cycle. By way of contrast, the unwanted edges corresponding to +a + c = 2a + d = x + d + 2 and b + e = 2b + d = x + d − 2 cannot be found +as vertex labels, because all vertex labels of the first K cycles (and also the +isolates) are divisible by four, including the label x + d. Finally, as all ‘new +labels’ are odd and all ‘old labels’ are even, an edge between an ‘old vertex’ +and a ‘new vertex’ must be labelled with a ‘new label’, and this also implies +that only the two bigger ‘new labels’ c and e could possibly serve as edge +labels. Moreover, as c is one congruent four, this must match the only other +label that is one congruent four, which is a, as all ‘old labels’ are divisible +by four. Hence, the question is if c − a = d is an ‘old label’, which is clearly +not the case by induction. Similarly, b and e are three congruent four, but +e − b = d and the same argument applies in this case as well. +□ +Notice that in the recursive labelling algorithm hidden in the previous +proof, the assumption that d does not occur as a vertex label is crucial, as +otherwise there would be an unwanted edge between a and d, because we +have the vertex label a + d. +In contrast to the labelling strategy described in the previous subsec- +tion, and in particular analyzed in Lemma 3, we obtain a worse relation +concerning the growth of the range for this new labelling strategy. +Lemma 5. There is a labelling strategy λ for disjoint unions of C4’s such +that rλ(G) ∈ O(2n/2) for a graph of order n which is a collection of n/4 +many C4’s. +13 + +Proof. Although also the size of the smallest label grows in this magni- +tude, it suffices to estimate the size of the largest label. For 2C4, this is +32 = 2 · 28/2, as described in Figure 5. As this largest label is always multi- +plied by 4 = 24/2, we get 2·2n/2 = 2·2(n−4)/2 ·24/2 by induction, considering +an n-vertex graph which is a collection of n/4 many C4’s. +□ +We can generalize the construction of Lemma 4 to get the following result. +Proposition 2. Let G be a graph with δ(G) = σ(G) = 2 such that there is +a sum-optimal labelling λ of the sum graph H = G + N2 such that λ(V (H)) +contains a NTAP, then there is a sum-optimal labelling λ′ of the sum graph +H′ = G + C4 + N2 such that λ(V (H′)) contains a NTAP. +For example, consider C3 + C4, starting with a labelling 1 − 3 − 4 of +the C3. Now, the isolates are 5 and 7. Observe that 1, 3, 5 is an arithmetic +progression whose offset 2 is not a vertex label. Hence, we can multiply the +numbers by 4 to get a labelling 4 − 12 − 16 of the C3, with isolate labels 20 +and 28. Finally, we label the C4 as 1 − 3 − 9 − 11. Clearly, the same isolate +labels of 20 and 28 suffice. +This proposition will come in handy when finally combining the results of +this subsection with that of the next one. The importance of the arithmetic +progression becomes also clear when revisiting Lemma 12. Unfortunately, +as already described in Lemma 5, the range will grow exponentially with +base +√ +2 if this proposition is applied repeatedly. +4.2. Collections of cycles without 4-cycles +We first discuss labelling strategies for collections of cycles without C4, +but Proposition 2 immediately shows a way how to add C4 afterwards, as +we explain in the following. +4.2.1. Dealing with long cycles +We will consider Fibonacci-labellings of cycles Cn, with n > 4,2 leaving +the case of triangles to be treated later. +λn(x, y) : (x, x + y, 2x + y, 3x + 2y, . . . , F(n)x + F(n − 1)y) +with isolates (F(n) + 1)x + F(n − 1)y and F(n + 1)x + F(n)y, using the +Fibonacci sequence F : (1, 1, 2, 3, 5, 8, 13, 21, . . .) with F(0) = 0 for conve- +nience. +2Collections of such longer cycles were treated in [13] in a similar fashion. +14 + +Lemma 6. For any n ≥ 3, n > 4, and any 1 ≤ x ≤ y, λn(x, y) gives a +sum labelling of Cn. +In fact, we can consider the labelling scheme λ(x, y)(k) = F(k)x+F(k−1)y. +Notice that the corresponding range function grows as ϕn for Cn, where ϕ +is the golden ratio number. +The parameters x and y (x < y) give us great flexibility. We could start +labelling the first of a certain number of longer cycles, starting with x1 = 1 +and y1 = 1. If the first cycle has length n1, the last of its vertices would be +labelled F(n1)x1 + F(n1 − 1)y = F(n1) + F(n1 − 1) = F(n1 + 1). The two +isolates (F(n1)+1)+F(n1−1) = F(n1+1)+1, F(n1+1)+F(n1) = F(n1+2), +giving us a new pair (x2, y2). If the second cycle has length n2, the last of +its vertices would be labelled F(n2)x2 + F(n2 − 1)y2 = F(n2)(F(n1 + 1) + +1) + F(n2 − 1)F(n1 + 2). We get similar expressions for the isolates, which +will again form the start (x3, y3) of labelling the next cycle etc. +We explain this labeling strategy by an example, a collection of three +C5 in Figure 8. The disadvantage of this strategy comes from the fact that +we do not see an arithmetic progression in it such that its offset is not a +vertex label. In particular, how do we label C5 + C4? +However, there is a remedy to it: Singla, Tiwari & Tripathi [10] showed +that (for rσ, the spum number) rσ(Cn) ∈ [2n − 2, 2n − 1] for n ≥ 4, and +rσ(Cn) = 2n − 1 for n ≥ 13. Namely, for odd n ≥ 5, they propose the label +set +L(Cn) = [n − 3, 2n − 4] ∪ {3n − 6, 3n − 4} +ordered as +(n − 3, n − 1, 2n − 5, n + 1, 2n − 7, n + 3, 2n − 9, . . . , 2n − 4, n − 2, n − 3) +For instance, this gives the labelling (2, 4, 5, 6, 3) of a C5, with isolates 9, 11. +Unfortunately, this is not a valid labelling as claimed in the paper, as we +get the unwanted edge between the vertices labelled 2 and 9, adding up +to the label 11. +The labelling works for n = 7, though, where we get +(4, 6, 9, 8, 7, 10, 5) with isolates 15, 17. Namely, we find that the difference +3n−6−(3n−4) = 2 between the two isolate labels only occurs in [n−3, 2n−4] +when n = 5. In [10], another labelling was proposed for even n ≥ 4:3 +L(Cn) = [n − 2, 2n − 3] ∪ {3n − 5, 3n − 3} +ordered like +(n − 2, 2n − 3, n, 2n − 5, n + 2, 2n − 7, . . . , 2n − 4, n − 1, n − 2) +3Other than claimed, the proposed labelling actually does not work for n = 4. +15 + +Thus, for n = 6, the C6 can be labelled (4, 9, 6, 7, 8, 5), with isolates 13, 15. +In all cases, we clearly find a NTAP, for instance for the proposed C6- +labelling 4, 5, 6 with offset 1. +As this is of importance for our algorithm, let us show that there does +indeed exist a sum labelling of the C5 that satisfies the conditions we need. +Lemma 7. The labelling (1, 2, 7, 9, 3), with isolates 4, 12, 16 contains the +arithmetic progression 2 − 7 − 12 whose offset 5 is not a label of any vertex. +This looks worse than what the Fibonacci labelling would deliver, as +we need three isolates, but it is in fact better, as 4, 12, 16 might be seen +as the start of a Fibonacci labelling on the second cycle. This proves that +σ(C5 + Cn) = 2 if n ̸= 4, and accordingly an optimal labelling can be +given that contains a NTAP. For instance, C5 + C3 can be labelled with +{1, 2, 3, 4, 7, 9, 12, 16}, with isolates {20, 28}. Only for the next cycles, we +employ the Fibonacci scheme. This preserves the property to have a NTAP. +Lemma 8. The labelling of C5 + C4 with the C5 labelled as (2, 4, 6, 10, 16) +and the C4 labelled as (1, 9, 18, 26) (with isolates 27, 44) contains the NTAP +10 − 27 − 44. +Proof. The C5-labelling follows a Fibonacci scheme. As 1 + 9 = 10, one +C4 edge label is in the C5, while its other edge labels are in the isolates. □ +This labelling scheme can be generalized as follows: +C5 : (a, b, a + b, a + 2b, 2a + 3b); +C4 : (c, 2b + c, 3a + 3b, 3a + 5b); +isolates : 3a + 5b + c, 6a + 8b. +The above labelling requires that a = 2c (to ensure that a+2b = 2b+2c +for the edge connecting 2b + c and c), and b ̸= 3c (to avoid the unwanted +edge a + (a + b) = b + 4c being represented by 2b + c). +To conclude this section, we show how to label 3C5 in two different ways. +Figure 8 shows a labelling were we strictly follow a Fibonacci scheme. If we +use Lemma 7 at the beginning (with the advantage of showing an arithmetic +progression as desired), we arrive at Figure 9. +16 + +1 +2 +3 +5 +8 +9 +13 +22 +35 +57 +66 +92 +158 +250 +408 +Figure 8: How to label a collection of cycles, here three C5’s, with isolates 474, 658. +1 +2 +7 +9 +3 +12 +16 +28 +44 +4 +48 +72 +120 +192 +312 +Figure 9: How to label a collection of cycles, here three C5’s, with isolates 360, 504, and +NTAP 2 − 7 − 12. +4.2.2. Dealing with triangles +We will prove the following assertion about collection of triangles (C3’s). +Lemma 9. Any Fibonacci labelling scheme for a non-empty collection of +triangles gives a valid sum labelling that contains a NTAP. +More precisely, consider +λ′ +n(x, y) : (x, x+y, 2x+y; 3x+y, 3x+2y, 6x+3y; 9x+4y, 9x+5y, 18x+9y; . . .) +as a labelling scheme serving for ℓ many C3’s, with isolates +3ℓx + ⌊3ℓ/2⌋y, +3ℓx + ⌈3ℓ/2⌉y . +This scheme is different in terms of growth compared to the schemes set up +before for longer cycles. Yet, it can be embedded in an inductive construc- +tion of a labelling of any number of cycles excluding C4. +To finally co-ordinate such a labelling with subsequently labelling a col- +lection of C4, we need to satisfy a NTAP. First, observe that y (in λ′ +n(x, y)) +is not the label of any vertex if y ̸= x. Then, in order to obtain an arith- +metic progression, we might find x+2y = 3x+y, or y = 2x. In other words, +we propose the labelling scheme +λ(3) +n (z) = λ′ +n(z, 2z) : (z, 3z, 4z; 5z, 7z, 12z; 17z, 19z, 36z; 53z, 55z, . . . ) +17 + +2 +4 +6 +10 +16 +18 +26 +1 +9 +27 +44 +71 +Figure 10: How to label C5 + C4 + C3, with isolates 98, 115, and NTAP 10 − 18 − 26. +for collection of triangles. For example, for a single C3, we can take the +labelling (1, 3, 4; 5, 7), with z = 1. +If we consider the sequence g(n) of +smallest labels per cycle in this sequence of labels, assuming z = 1, we get +the recursion g(1) = 1 and g(k) = 3g(k − 1) + 2. This proves that g(k) ∈ +Ω(3k). Hence, the range of the labelling scheme λ(3) +n (z) grows exponentially, +similar to ( +3√ +3)n with the number n of vertices. +We can hence use these schemes to label any collection of cycles without +C4, simply by interpreting the two isolates ι1 and ι2 necessary to label a +collection of ℓ cycles (by induction hypothesis) as the first two vertices, +say, x and x + y, of the next cycle. Also, it is clear that any collection of +cycles that contains at least one triangle and that is labelled this way has +a NTAP. Hence, we can apply Proposition 2 to add a collection of C4 on +top. At each time, we only need two isolates for this collection of cycles, +apart from one exception, when the whole collection only contains one C5, +where a special labelling was described in Lemma 8. This proves our main +theorem for 2-regular graphs. +Although our labelling strategy λ for 2-regular graphs G attains σ(G), +one can see that rσ(λ) ∈ O(ϕn) in the worst case, where ϕ is the golden +ratio number. But if the cycle collection contains only smaller cycles, the +growth rate becomes smaller. Nonetheless, it stays exponential, and it is +unclear if there are labelling strategies for 2-regular graphs whose range +stays polynomial. +As a final comment, notice that the sequence of labellings that we pro- +pose, i.e., our labelling strategy, is not the only possible one, as shown in +Figure 10, where the labelling of C5 + C4 as shown in Lemma 8 is finally +combined with labelling a C3. Our standard labelling strategy would be a +bit worse, as shown in Figure 11. +18 + +4 +8 +28 +36 +12 +5 +3 +25 +23 +16 +48 +64 +Figure 11: Labelling C5 + C3 + C4, with isolates 124, 140, and NTAP 8 − 28 − 48. +5. Bringing paths into the game +We are first discussing a general situation that we face after having dealt +with all cycles. Here, we have to distinguish two cases: either, this cycle +collection has sum number two, or it has sum number three, which means, +it is a single C4. +5.1. Dealing with cycle collections of sum number two +Proposition 3. Let G = (V, E) be a graph with σ(G) = 2, testified by a +labelling λ with isolate labels ι1 and ι2. If ι1 + ι2 ̸= λ(u) + λ(v) for any two +vertices u, v ∈ V , then σ(G + Pk) = 1 for any k ≥ 2. +Proof. Assume ι1 < ι2. Recall the Fibonacci labelling scheme for paths +(Equation (2)). We propose to use λφ +ι1,ι2 to label Pk. Clearly, ι2 and ι1 + ι2 +are the two biggest labels, labelling the second and third vertex on the path +(or the isolate if k = 2). This is an invariant that is maintained by the +Fibonacci scheme: the labels of the ℓth and (ℓ + 1)th vertex on the path are +always greater than any previous labels. By this and due to the assumption +that ι1+ι2 ̸= λ(u)+λ(v) for any two vertices u, v ∈ V , this labelling cannot +introduce unwanted edges and is hence valid for G + P. Moreover, it will +leave us with one isolate only. As δ(G + P) = 1, the labelling is optimal. □ +Notice that the argument also works if σ(G) > 2; just pick the largest +isolate label plus any other isolate label to produce the label λ3 (or ι if the +added path has length two). This proves the following fact: +Corollary 1. Let G = (V, E) be a graph with σ(G) ≥ 2, testified by a +labelling λ with largest isolate labels ι1 and ι2. If ι1 + ι2 ̸= λ(u) + λ(v) for +any two vertices u, v ∈ V , then σ(G + Pk) ≤ σ(G) − 1 for any k ≥ 2. +19 + +Notice that this argument is different from the (more general) one pre- +sented in [14] where σ(G1 + G2) ≤ σ(G1) + σ(G2) − 1 is proved under the +assumption that optimum labellings λ1 of G1 and λ2 of G2 exist such that +there is an element in λi(V (Gi)) that is relatively prime to the largest ele- +ment of λ3−i(V (G3−i)) for i = 1 or i = 2. Also, observe that the labels will +grow exponentially by a Fibonacci labelling scheme. +5.2. Combining a 4-cycle with paths +Given the ideas of Lemma 4 and the results so far, one might be tempted +to think that the sum number of every graph G of maximum degree 2 with +a disjoint copy of C4 is equal to the minimum degree of G. Our next result +shows this is not the case. +Proposition 4. σ(C4 + P2) = 2. +Before proving Proposition 4, we require some observations about C4. +These observations also provide an indication as to why C4 is different from +all the other cycles (as far as the sum number is concerned, at least). +It was already shown by Harary [2] that σ(C4) = 3. Here we present a +reason for this in the following, as it indicates the way how lower bounds +on σ can be shown when the minimum degree criterion (proving σ(Cr) ≥ +δ(Cr) = 2 for each r ≥ 3) is insufficient. +Lemma 10. Let C4 + G be a graph without isolates, and let H be a sum +graph of C4 + G. Then, all vertices corresponding to edge sums of the C4 +lie in H − C4. +In particular, this means that every sum labelling of a C4 is exclusive,4 +and that hence σ(C4) = 3 holds because of the next lemma (Lemma 11). +Proof (Proof of Lemma 10). We will prove this by contradiction. Let +the vertex labels of the C4 be (a, b, c, d) in cyclic order. Assume to contrary +that a + b = c (due to the symmetry of C4, all other cases are similar). +Then, we claim that all of the following are true. +(i) There is a vertex labelled b in H. +(ii) There is a vertex labelled a + d in H. +4This means that edge labels are among the isolate labels. +20 + +(iii) There is a vertex labelled a + b + d in H. +Since b lies in C4, (i) is true. Since (a, d) is an edge and H is a sum +graph, (ii) is true. Finally, since (c, d) is an edge, H is a sum graph, and +c+d = a+b+d, (iii) is also true. Now, since H is a sum graph, (i), (ii), (iii) +together imply that there must be an edge between the vertex labelled b +and the vertex labelled a + d. But b has exactly two neighbours, labelled a +and c, so a + d must be one of them. We will show that either case leads to +a contradiction. +If a = a + d, then d = 0, which is impossible, as H is a sum graph. If +c = a + d, then c = a + b implies that b = d, which is impossible, as H is a +sum graph. +□ +Lemma 11. Let C4 + G be a graph without isolates, and let H be a sum +graph of C4 + G. Let S be the set of numbers that correspond to the four +edge sums of the C4. Then, |S| ≥ 3. +Proof. We will show that |S| ≤ 2 leads to a contradiction. That is, the +four edges of the C4 must have at least three distinct edge sums. Let the +vertex labels of the C4 be (a, b, c, d) in cyclic order. Two edges that share +a vertex cannot have the same edge sum, because then there would be two +vertices with the same label. Thus, the only way that the C4 can have only +two distinct edge sums is if both the following hold. +a + b = c + d +a + d = c + b. +Subtracting the first equation from the second, we obtain that b−d = d−b, +or b = d, which is impossible in a sum graph. This completes the proof. □ +Lemma 12. Let C4 + G be a graph without isolates, and let H be a sum +graph of C4+G. Let S be the set of numbers that correspond to the four edge +sums of the C4. If |S| = 3, then the three numbers in S are in arithmetic +progression. +Proof. Let (a, b, c, d) be a labelling of the C4 such that a + b = c + d. Let +sum = a + b = c + d; +diff = c − a = b − d. +21 + +We will show that the labels of the three isolates are +iso1 = sum − diff; +iso2 = sum; +iso3 = sum + diff. +The labels of the edges (a, b) and (c, d) are equal to iso2, due to the definition +of sum. As for the labels of the edges (a, d) and (b, c), we have the following. +a + d = (c + d) − (c − a) = sum − diff = iso1; +b + c = (a + b) + (c − a) = sum + diff = iso3. +Since iso1, iso2, iso3 are clearly in arithmetic progression, this completes the +proof of Lemma 12. +□ +Finally, we are ready to prove Proposition 4. +Proof (Proof of Proposition 4). Label the C4 as (1, 7, 13, 19), the P2 +as (20, 32), and the two isolates as 8 and 44. It is easy to check that this is +a sum graph, and thus σ(C4 + P2) ≤ 2. +To prove that σ(C4 + P2) ≥ 2, assume to contrary that σ(C4 + P2) = 1. +Let the labels of the vertices of the P2 be (b1, b2) (assume b1 < b2), and the +isolate be b3. Recall that every C4 has at least three distinct edge labels +(Lemma 11), and none of those labels can be present in the vertices of the +C4 itself (Lemma 10). Thus, the only option is that the edge labels of the C4 +are b1, b2, b3. Furthermore, we know that whenever C4 has exactly three edge +labels, those three numbers form an arithmetic progression (Lemma 12). +Now, observe that the largest label of P2 (namely, b2) must be the largest +label of the graph G = C4 + P2, since G has only one isolate. Thus, for the +edge label b1 + b2 of the P2, we have: +b1 + b2 = b3. +(3) +As mentioned in the previous paragraph, since b1 < b2 < b3 are the three +edge labels of the C4, they are in arithmetic progression, implying that +b3 − b2 = b2 − b1. +(4) +With Equation (3) and Equation (4), we get b2 = 2b1 and b3 = 3b1, or +bi = ib1 +∀ i ∈ {1, 2, 3}. +(5) +22 + +Consider a P3 subgraph of the C4 such that one of the two edges of the P3 is +labelled b2. More precisely, let the P3 be (a1, a2, a3) such that a1 + a2 = b2. +Since a1 ̸= a3, the edge (a2, a3) cannot be labelled b2, too. Thus, (a2, a3) +is labelled either b1, or b3. That is, a2 + a3 is equal to either b1, or b3. If +a2 + a3 = b1, then +a1 − a3 = (a1 + a2) − (a2 + a3) += b2 − b1 += 2b1 − b1 +Using (5) += b1. +If a2 + a3 = b3, then +a3 − a1 = (a2 + a3) − (a1 + a2) += b3 − b2 += 3b1 − 2b1 +Using (5) += b1. +Therefore, either a1 = b1 +a3, or a3 = b1 +a1. In other words, either (b1, a3) +is an edge, or (b1, a1) is an edge. In either case, there is an edge between +the C4 and the P2, which is a contradiction because they are supposed to +be disjoint. +□ +On the other side, we can prove: +Lemma 13. σ(C4 + Pk) = 1 for all k ≥ 3. +Proof. If k = 3, label the C4 as {1, 3, 9, 11} and the P3 as {12, 4, 16}, with +the isolate being 20. If k ≥ 4, then the first four labels of the path Pk = +(v1, v2 . . . , vk) are λ(v1) = 12, λ(v2) = 4, λ(v3) = 16, λ(v4) = 20. After that, +we simply continue in the Fibonacci fashion, i.e., λ(vi+1) = λ(vi) + λ(vi−1), +with the label of the isolate being λ(vk) + λ(vk−1). It is easy to check that +no unwanted edges are introduced. +□ +The general algebraic strategy can be best seen by labelling the C4 +with {1, 3, 9, 11}, with a < b being the smallest numbers. We assume that +a + d = b + c, i.e., d = b + c − a. Moreover, we label the three path vertices +with {a+d, a+b, (a+d)+(a+b) = a+2b+c}. In order to save on isolates, +we also require that c+d = b+2c−a equals (a+2b+c)+(a+b) = 2a+3b+c, +23 + +which implies c = 3a + 2b. In summary, given small numbers a < b, we +construct the further labels of C4 as 3a + 2b and 2a + 3b. Then, the labels +on the path would be 3(a + b), a + b, 4(a + b), with the isolate 5(a + b). +If we want to label C4 + Pk with k ≥ 4, it is possible to save on the size +of the labels by starting with labelling the C4 with {1, 2, 6, 11} and the P4 +with {17, 3, 8, 12}, plus one isolate labeled 20. Further savings are possible +if we label the C4 with {2, 5, 8, 11}, as done as a standard throughout this +paper. The first five labels of the path Pk, k ≥ 5, with vertices v1, . . . , vk are +then: λ(v1) = 26, λ(v2) = 13, λ(v3) = 7, λ(v4) = 19, λ(v5) = 20. If k = 5, +then 39 would be the label of the isolate. Otherwise, we just continue in a +Fibonacci-style, i.e., λ(vi+1) = λ(vi) + λ(vi−1), with λ(vk) + λ(vk−1) being +the isolate. Again, no unwanted edges are introduced. +There is only one case left over to complete the picture: +Lemma 14. σ(C4 + 2P2) = 1. +Proof. By Proposition 4, σ(C4 + P2) = 2. +The labelling satisfies the +requirements of Proposition 3, which shows the claim. +□ +5.3. Combining cycles with more than one path +The following proposition also covers the case of pure path collections. +Proposition 5. Let G be a graph with σ(G) = 1. Then, σ(G + Pk) = 1 for +any k ≥ 2. +Proof. Let ι be the label of the isolate of a labelling λ of G certifying its +sum number to be 1. Then, ι is bigger than any vertex label of G. There- +fore, labelling Pk with the Fibonacci labelling scheme λφ +ι,2ι, as introduced in +Equation (2) in general form, labels G+Pk (together with λ) with only one +isolate, not creating conflicts, as all edge labels of G are smaller than 2ι. □ +We already saw (or will see soon) that a cycle collection plus one path +has sum number one with one exception, which is C4+P2. As we will fix the +only remaining case of C4 + 2P2 separately in Lemma 14, we can conclude: +Proposition 6. Let C be a collection of cycles and P be a collection of at +least two paths. Then, σ(C + P) = 1. +Proof. Except for the case of C4+P2, we know that σ(C +Pk) = 1 for any +collection of cycles C and any k ≥ 3. As we consider paths in decreasing +length, we will pick a cycle of length at least three to be considered first +if there is any. Therefore, if P is a collection of at least two paths, then +we can conclude σ(C + P) = 1 either by Proposition 5 directly, or by first +taking Lemma 14. +□ +24 + +6. Conclusion and Open Problems +We have explained that the labelling of a C5 as proposed in [10] is not +working correctly. +This leaves the spum-minimization problem open for +this particular small graph. But this question easily generalizes to nearly +all graphs with maximum degree of at most two as discussed in this paper. +In most cases, we only found labellings with labels of exponential size. This +might be necessary, but for such statements, we do not have any proof idea. +Our main result concerns the sum number of (all) graphs of maximum +degree two. Kratochv´ıl, Miller and Nguyen posed in [3] two conjectures that +are tightly related to our paper; we will formulate them as questions below. +• Given two graphs G1, G2 with σ(G1) = σ(G2) = 1, is it true that the +sum number of their disjoint union is always one? +• More generally: given two graphs G1, G2, is it true that σ(G1 +G2) ≤ +σ(G1) + σ(G2) − 1? +Observe that we did resolve the first question if G2 is a path (Proposition 5), +but the general question is still open. Upon some thought, it can be seen +that the general question is related to the following natural combinatorial +question: Find a characterization of all graphs with sum number one, also +known as unit graphs in the literature. We also refer to a recent paper [15] +that studies variations of this question. Finally, a slightly weaker but more +structured notion of sum labelling (called arithmetic graphs) could lend +some ideas that might help in resolving this question (and more optimisti- +cally, the second question) [16]. +Apart from these combinatorial questions, the basic complexity ques- +tions concerning the graph parameter σ and rσ are open. For instance, is +it NP-hard to decide if, given a graph G and a number k, σ(G) ≤ k holds? +One of our own motivations to study graphs of maximum degree two was +to see if one could use the operation of graph union to piece gadgets to- +gether for this and similar questions. But we are still far from this, as even +these seemingly easy questions concerning graphs of maximum degree two +are non-trivial to solve. +References +[1] J. A. Gallian, A dynamic survey of graph labeling, version 23, The Electronic Journal +of Combinatorics DS 6 (2020). +25 + +[2] F. Harary, Sum graphs and difference graphs, Congressus Numerantium 72 (1990) +101–108. +[3] J. Kratochv´ıl, M. Miller, H. M. Nguyen, Sum graph labels – an upper bound and +related problems, in: 12th Australasian Workshop on Combinatorial Algorithms, +AWOCA, Institut Teknologi Bandung, Indonesia, 2001, pp. 126–131. +[4] W. F. Smyth, Sum graphs of small sum number, Colloquia mathematica Societatis +J´anos Bolyai 60 (1991) 669–678. +[5] J. Ryan, Exclusive sum labeling of graphs: A survey, AKCE International Journal +of Graphs and Combinatorics 6 (1) (2009) 113–136. +[6] H. Nagamochi, M. Miller, Slamin, On the number of isolates in graph labeling, +Discrete Mathematics 243 (2001) 175–185. +[7] N. Hartsfield, W. F. Smyth, A family of sparse graphs of large sum number, Discrete +Mathematics 141 (1-3) (1995) 163–171. +[8] M. Sutton, M. Miller, On the sum number of wheels, Discrete Mathematics 232 +(2001) 185–188. +[9] H. Fernau, K. Gajjar, The space complexity of sum labelling, in: +E. Bampis, +A. Pagourtzis (Eds.), Fundamentals of Computation Theory - 23rd International +Symposium, FCT, Vol. 12867 of LNCS, Springer, 2021, pp. 230–244. +[10] S. Singla, A. Tiwari, A. Tripathi, Some results on the spum and the integral spum +of graphs, Discrete Mathematics 344 (5) (2021) 112311. +[11] M. N. Ellingham, Sum graphs from trees, Ars Combinatoria 35 (1993) 335–349. +[12] T. Hao, On sum graphs, Journal of Combinatorial Mathematics and Combinatorial +Computing 6 (1989) 207–212. +[13] H. Burhan, R. Rusin, K. A. Sugeng, Optimum sum labeling of finite union of sum +graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 65 +(2008) 133–138. +[14] M. Miller, J. Ryan, W. F. Smyth, The sum number of a disjoint union of graphs, +in: 14th Australasian Workshop on Combinatorial Algorithms (AWOCA), Seoul +National University, Korea, 2003, pp. 120–124. +[15] M. Koneˇcn´y, S. Kuˇcera, J. Novotn´a, J. Pek´arek, ˇS. ˇSimsa, M. T¨opfer, Minimal sum +labeling of graphs, Journal of Discrete Algorithms 52-53 (2018) 29–37. +[16] B. D. Acharya, S. M. Hegde, Arithmetic graphs, JGTh 14 (3) (1989) 275–299. +26 + diff --git a/_9A0T4oBgHgl3EQfPv8L/content/tmp_files/load_file.txt b/_9A0T4oBgHgl3EQfPv8L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbecb937b174dbfe4db9857470278186523e1c71 --- /dev/null +++ b/_9A0T4oBgHgl3EQfPv8L/content/tmp_files/load_file.txt @@ -0,0 +1,679 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf,len=678 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='02178v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='CO] 5 Jan 2023 Sum Labelling Graphs of Maximum Degree Two Henning Fernau (Universit¨at Trier, Universit¨atsring 15, Trier, Germany)1, Kshitij Gajjar (Indian Institute of Technology Jodhpur, Rajasthan, India)1 Abstract The concept of sum labelling was introduced in 1990 by Harary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A graph is a sum graph if its vertices can be labelled by distinct positive integers in such a way that two vertices are connected by an edge if and only if the sum of their labels is the label of another vertex in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is easy to see that every sum graph has at least one isolated vertex, and every graph can be made a sum graph by adding at most n2 isolated vertices to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The minimum number of isolated vertices that need to be added to a graph to make it a sum graph is called the sum number of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The sum number of several prominent graph classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', cycles, trees, complete graphs) is already well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We examine the effect of taking the disjoint union of graphs on the sum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In particular, we provide a complete characterization of the sum number of graphs of maximum degree two, since every such graph is the disjoint union of paths and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Keywords: Sum labelling, Sum number, Cycles, Paths, Graph union 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Introduction The area of graph labelling is a specific subarea of graph theory that has developed an enormous body of literature, as testified by Gallian’s dynamic survey [1] which mentions over 3000 research papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' One of these labellings is sum labelling, introduced by Harary [2] as a form of representing graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is known [3] that every n-vertex graph G can be represented via a sum labelling, which means that it is possible to add at most n2 isolated vertices (also called isolates, in short) to G to make it a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This makes sum Email addresses: fernau@uni-trier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='de (Henning Fernau (Universit¨at Trier, Universit¨atsring 15, Trier, Germany)), kshitij@iitj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='in (Kshitij Gajjar (Indian Institute of Technology Jodhpur, Rajasthan, India)) Preprint submitted to Discrete Mathematics January 6, 2023 1 3 2 5 4 1 4 3 7 5 (a) (b) (c) Figure 1: (a) This graph is not a sum graph, because it has no isolated vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (b) This is an incorrect sum labelling of a sum graph, because (1, 4) is not an edge yet there is a vertex labelled 1 + 4 = 5 in the graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (c) This is a correct sum labelling of a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' labelling a compelling concept from the viewpoint of computer science also, because it may be that certain graphs can be encoded much more succinctly with sum labellings than with the more traditional ways of storing graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let us now fix some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We deal with simple, undirected graphs, specified (as usual) as G = (V, E), where V is the (finite) set of vertices of G, and E is its set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If v is an endpoint of an edge e, then we say that v and e are incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The number of edges incident to a vertex is the degree of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let N denote the set of all natural numbers (positive integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, we say that G is a sum graph if there exists an injective mapping λ : V → N (called the sum labelling of the vertices of G) such that E = {xy | ∃z ∈ V : λ(z) = λ(x) + λ(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Up to isomorphism, the set of numbers λ(V ) therefore determines G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In other words, λ encodes G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As isolated vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', vertices of degree zero) are usually irrelevant in applications, λ(V ) can be viewed as the description of G\\I, where I is the set of all isolated vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, λ(V ) is called the sum number encoding of G \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Conversely, given a graph G without isolates, the minimum number of isolates that need to be added to G in order to make it a sum graph is called the sum number of G, written as σ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, G + Nσ(G) is a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (Here, + denotes the disjoint union of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, Ni denotes the null graph (edgeless graph) on i vertices, or equivalently, a set of i isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=') See Figure 1 for some examples and non-examples of sum graphs and sum labellings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A labelling function λ can be also seen as operating on edges by the summability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' λ(e) for an edge e = xy ∈ E is defined as λ(x)+λ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, though only the vertices are labelled by a sum-labelling, we sometimes 2 also refer to its edges as labelled by the sum of its endpoints (two different edges can have the same edge label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Are substantial savings possible with sum number encodings of graphs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Some partial answers are possible from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For instance, σ(Kn) = 2n−3 is known for n ≥ 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', 3n−3 numbers suffice to store the information about the complete graph Kn, while traditional methods would need O(n2) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As mentioned in [4], this can be obtained by labelling vertex xi with 4i − 3, with 1 ≤ i ≤ n, leading to isolate labels 4j + 2 for 1 ≤ j ≤ 2n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, the sizes of the labels are in fact linear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The focus of our study is the sum number of certain graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This follows much of the tradition in the literature, as can be seen in surveys like [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More precisely, we prove as our main result a complete picture of the sum number of every graph of maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As a consequence, if G has maximum degree two, then σ(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This is not completely expected, as it is known that the sum number of general graphs grows with the number of edges [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In fact, this can happen even with sparse graphs [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' When talking about sum labelling a whole infinite family of graphs G, often with the additional property that for each positive integer n, there is at most one graph Gn of order n within G, we also speak of a labelling scheme λ : N → N that formalizes the labelling strategy that we suggest for G in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For Gn, to be labelled with i isolates, we take {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , n} as the vertex set of Gn and consider the set of numbers {λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , λ(n+ i)} as the set of labels of the sum graph Gn +Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Extending this notion, a general labelling scheme is specified by three functions λ : N → N, σ : N → N and ι : N → N that are interpreted as a labelling strategy for Gn ∈ G of order n, with i isolates, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As the set of vertices of Gn + Ni, we consider Vn+i = {σ(n), σ(n) + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , σ(n) + n − 1, ι(n), ι(n) + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , ι(n) + i − 1}, where the first n numbers denote the vertices of Gn, and as labels we take λ(j) with j ∈ Vn+i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This boils down to a labelling scheme if σ(n) is constant one and ι(n) = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More general labelling strategies of n-vertex graphs of a family of graphs G are possible and will be discussed later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Our main result is a complete precise characterization of all graphs G of maximum degree two: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G be a graph of maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, σ(G) = δ(G) except for two graphs, namely C4 and C4 + P2, for which σ(G) = δ(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Harary [2] already showed that σ(C4) = 3, and that the minimum degree of a graph is always a lower bound on its sum number (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', σ(G) ≥ δ(G) 3 Sum-labelling graphs of maximum degree two: a strategy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Firstly, we deal with all cycles of length not equal to four (if any), in descending order of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Secondly, we deal with all cycles of length four (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, we deal with paths (if any), in descending order of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Figure 2: Our proposed strategy for sum-labelling graphs G with 1 ≤ δ(G) ≤ ∆(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' for all graphs G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, to prove our main theorem, it suffices to show that σ(G) ≤ δ(G) for all graphs G of maximum degree two, except for C4 and C4 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' An additional proof is required to show that σ(C4 + P2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Apart from having a combinatorial result, we can also interpret our proof as providing an algorithm that labels any graph of maximum degree two optimally with respect to its sum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For the motivation of efficiently storing graphs, this is not completely satisfying, as the sizes of the labels could be exponential in the number of vertices of the graph according to our constructions, which means that we might need up to O(n2) many bits for storing an n-vertex graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In principle and in general, we can do this more efficiently in terms of label sizes [9], but the algorithm presented in [9] is not tailored towards using as few isolates as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', it does not obey the sum number of the graph, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Notice that every graph of maximum degree two is a disjoint union of cycles and paths (in other words, each connected component of the graph is either a path or a cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To prove our main theorem, we will deal with the connected components in a specific sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This naturally produces an algorithm that optimally labels (with respect to the sum number) all graphs with maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We provide a sketch of our strategy in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Figure 2 also explains the sequence in which we will treat all graphs of maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For example, if the graph G is G = 5C3 + 2C4 + C6 + 2C7 + 3C9 + 4P2 + P5 + 2P8 + P9, then we will deal with the components of G in the following order: 3C9, 2C7, C6, 5C3, 2C4, P9, 2P8, P5, 4P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The space complexity of sum labelling One of our motivations to return to sum labellings was the idea that one can use them to efficiently store graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This idea was already expressed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There, they consider the notion of the range r(λ) of a labelling λ, which is defined as the difference between max λ(V ) and min λ(V ),1 with r(λ) = max v,v′∈V λ(v) − λ(v′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To clearly distinguish our notion of range from the ones mentioned in footnote 1, let us introduce the sum range number rσ(G) of a graph G as the smallest range of a labelling of a sum graph G + Nk for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As eventually the range grows with the number of vertices, here we propose two different ways of ensuring that the numbers involved do not grow too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To better motivate the introduction of these new graph parameters, let us first analyze the sizes needed to store graphs in a database using a sum labelling encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A graph G = (V, E) on n vertices can be stored as follows: We need O(log n) bits to store n itself, plus O(log log(max λ(V ))) bits to store log2(max λ(V )), O(log σ(G)) bits to store the number of isolates and then log2(2 max λ(V )) · (n + σ(G)) more bits for the (at best ordered) list of numbers (vertex and isolate labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In the end, we have to store a list of n + σ(G) many integers, each with log2(2k) many bits, because edge labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', labels of isolates) have value of at most 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Instead, one could also first store the smallest label and then one would only need log2(r) bits per number, where r is the range of the labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More precisely, if we want to given an estimate of the number of bits needed to store graph G = (V, E) with the labelling λ, we get the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 2(log2 n + min v∈V log2(λ(v))) + |λ(V ∪ E)| · log2(r(λ)) (1) Notice that although it looks beneficial to minimize |λ(V ∪ E)| by choosing a labelling λσ that achieves σ(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', where |λσ(V ∪ E)| = |V | + σ(G), there could be another labelling λ with |λ(V ∪ E)| > |V | + σ(G), but r(λ) could be much smaller than r(λσ), potentially out-weighing the disadvan- tage of needing more isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This is true in particular when r takes values exponential in n, as for the Ellingham-labelling for trees [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 1 In [3] and also in [10], under the name spum, the mentioned difference is considered only for labellings that attain the sum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5 Further stretching our notation, we will also consider rλ for a labelling strategy λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', for a way to label a whole family of sparse graphs as de- scribed above, so that rλ can be viewed as a mapping that associates to n the largest range of any labelling of an n-vertex graph according to this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, we can analyze the growth of rλ for certain labelling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' What is the main purpose of a graph database?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Clearly, one has to access the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A basic operation would be to answer the query if there is an edge between two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Now, if max λ(V ) is polynomial in n = |V |, we can answer this query in time O(log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Namely, assuming the polynomial bound on the size of the labels, we would need time O(log(n)) to add the two labels of the vertices, and we also need time O(log(n)) to search for the sum in the ordered list of numbers, using binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Otherwise, the additional time O(log(max λ(V ))) would be quite expensive, probably making the idea of storing large graphs as sum graphs in databases unattractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, also the range of labellings should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Other parameters that measure the space consumption of storing graphs even more accurately have been discussed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' However, for the discus- sions in this paper, the two parameters λ and r(λ) suffice, also because these are more accessible from the combinatorial viewpoint that we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The main difficulty in dealing with the combinatorics of sum labelling prevails also for these modified definitions, which is the question of how to prove lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The only general assertion that is available is to say that the sum number of a graph is at least as big as its minimum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There are also generalizations of this observation based on degree sequences (see [12, 4]), but this is irrelevant to us, as we consider graphs of bounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For instance, this means that the sum number of a collection of cycles is at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' But, as we see in the following, even proving that certain collections of cycles have a sum number of two is far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There are no really systematic tools available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Regarding the notion of sum range number, it is nice to observe that the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1 of [10] concerning the spum of a graph is also valid in our case (which is, as discussed above, a definitorial variation of spum), so that we can state without proof the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G be a graph of order n with minimum degree δ(G) and maximum degree ∆(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, rσ(G) ≥ 2n − (∆(G) − δ(G)) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Observe that for regular graphs, the lower bound stated in the previous proposition simplifies to 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Unfortunately, even for our simple graph families, we reach this bound only occasionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 6 16 31 17 30 18 29 19 28 20 27 21 26 22 25 23 24 47 (b) (a) 2 3 5 6 11 12 23 24 47 48 95 96 191 192 383 384 767 Figure 3: (a) The exponential labelling scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (b) The linear labelling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A first example: labelling a disjoint collection of edges This section should be treated as an introductory example into the intri- cacies of sum labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It has also been studied earlier [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, it covers an important subcase of our main theorem, which is 1-regular graphs, or graphs of (maximum) degree one (without isolates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, one can see examples that deal with the union of two graphs, each of sum number one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is known that all trees have sum number 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' according to a remark following Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1 in [11], all forests also have sum number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' However, it is not that clear how fast the label sizes grow in these constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, recall that it is still an open question for general graphs with sum number one whether their graph union again has sum number one [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, we will present two different constructions that label a disjoint collection of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More mathematically speaking, we will show two labelling schemes for the family of 1-regular graphs: an exponential labelling and a linear labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' An exponential solution If you have n vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', n/2 edges), label the first edge as (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The second edge starts with the edge label of the first edge (2 + 3 = 5, so the second edge is labelled (5, 6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The third edge starts with the edge label of the second edge (5 + 6 = 11, so the third edge is labelled (11, 12)), and so on (see Figure 3 (a) for an example with n = 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Generalising this, the following labelling scheme λ : N → N works for 7 every 1-regular graph: λ(n) = \uf8f1 \uf8f2 \uf8f3 2 if n = 1 λ(n − 1) + 1 if n is even λ(n − 2) + λ(n − 1) if n is odd and n > 1 The Online Encyclopedia of Integer Sequences suggests that this is an- other variation on Ulam numbers if we think of the starting point to be λ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, λ(n) (for n > 1) can be seen as the smallest (when n is even) or largest (when n is odd) number bigger than λ(n − 1) that is a unique sum of two distinct earlier terms of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This connection also suggests the following closed form: λ(n) = � 3 · 2k−1 if n is even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', n = 2k 3 · 2k − 1 if n is odd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', n = 2k + 1 In other words, we have λ(n) ∈ Θ ��√ 2 �n� , implying that it is exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Although the suggested labelling λ is optimal with respect to the sum number σ, we see: r(λn) ∈ Θ ��√ 2 �n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Can we do better with respect to the sum range number?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A linear solution Consider the following general labelling scheme for 1-regular graphs (ob- serve that n is necessarily even) that we first describe in a more intuitive fashion, already indicating the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (n, 2n − 1), (n + 1, 2n − 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , �3n 2 − 1, 3n 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Here, writing (λ(u), λ(v)) refers to two vertices u, v that are connected by an edge (see Figure 3 (b) for an example with n = 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Notice that all edge labels sum to 3n − 1 (which is the isolate), and even the sum of the two smallest labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', n + (n + 1) = 2n + 1, is smaller than 3n − 1 but bigger than any other label in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More formally, we consider the functions λ, σ, ι with λ(n) = σ(n) = n and ι(n) = 3n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This gives as vertex names {n, n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , 2n − 1} for a 1-regular graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This general labelling scheme can be further generalized by using the parameters (x, y, d, k), with x < y (in our example, x = n, y = 2n − 1, d = 1, k = n/2 − 1), by putting (x, y), (x + d, y − d), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , (x + kd, y − kd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 8 All labels sum up to x + y, which is the isolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As long as the sum of the two smallest labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', 2x+d, is smaller than x+y but bigger than y, such a sum labelling is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As the scheme consists of interleaving an increasing arithmetic progression with a decreasing arithmetic progression (with the same “slope”), we call such schemes arithmetic progression schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The concrete arithmetic progression scheme that we first suggested has as its range the numbers n through 2n−1 and is hence (nearly) optimal, as Proposition 1 gives 2n − 2 as a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Singla, Tiwari & Tripathi [10] show an upper bound of 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, we know that the optimal answer is either 2n − 2 or 2n − 1, but we do not know which one it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Labelling paths The ideas presented for 1-regular graphs work for paths also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As we will need the exponential labelling scheme explicitly in the following, we are going to present (only) this one now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For the linear solutions, we refer to [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A scheme could be based on fixing two positive integers x, y as parame- ters, and then defining the labelling scheme λφ x,y : N → N as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' λφ x,y(n) = \uf8f1 \uf8f2 \uf8f3 x if n = 1 y if n = 2 λφ x,y(n − 2) + λφ x,y(n − 1) if n > 2 (2) Due to the similarity to Fibonacci numbers, it is clear that λφ x,y(n) = O(φn), where φ is the golden ratio number, irrespectively of the start values x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We can hence deduce the following well-known fact by this Fibonacci scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For any n ∈ N, σ(Pn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Several labelling strategies for collections of cycles Recall that according to the algorithmic strategy sketched in Figure 2, we first deal with all cycles of length five and larger, then with all triangles, and finally with all cycles of length four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The collection of C4 is the most tricky one, as it could possibly leave us with three intermediate isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Apart from this special situation, we will always face the situation that after having dealt with k − 1 cycles, we have two isolates that we integrate into the kth cycle as the start of a new Fibonacci-type labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This is discussed in detail in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 9 For the inductive argument, it becomes crucial to know that our la- belling contains a non-trivial arithmetic progression, or NTAP for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This means that we find three labels x, x + d, x + 2d in the proposed la- belling such that the offset d is not a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Collections of 4-cycles In this subsection, we actually present two labelling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The first one could be called “linear-exponential” in the sense that the proposed labelling strategy is linear (an arithmetic progression) per cycle, but from cycle to cycle, we observe an exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It uses three isolates (always) but has a smaller range compared to the second strategy that uses two isolates only (from two C4 onwards) but needs a larger range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' A linear-exponential labelling scheme Consider the labelling (2, 5, 8, 11) of a C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Notice that the progression is arithmetic, with a difference of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' All numbers are congruent 2 modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The three isolates are: (7, 13, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This arithmetic progression, with a difference of 6, can be again lifted to a labelling of a second C4, which is then (7, 13, 19, 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' All numbers are congruent 1 modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The three isolates are now: (20, 32, 44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This arithmetic progression, with a difference of 12, can be again lifted to a labelling of a third C4, which is then (20, 32, 44, 56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' All numbers are congruent 2 modulo 3, as with the first C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is clear that we can continue this construction by adding a fourth C4 with labels (52, 76, 100, 124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' All numbers are congruent 1 modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To wrap up, the odd-numbered cycles get numbers that are congruent to 1 modulo 3, while the even-numbered cycles get numbers which are con- gruent to 2 modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' These modulo 3 observations show that no edges can ever occur between vertices in subsequent cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As all the edge labels of the ith cycle can be found on the (i + 1)th cycle, we can see that (as the differences on the ith cycle are of the form 3 · 2i−1), the non-edges (diag- onals) on the ith cycle cannot be represented by vertices on the (i + 1)th cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' By the aforementioned exponential growth of the labels one cycle to the next cycle, further non-edges cannot be represented by the suggested numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proves: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If G is a disjoint union of C4’s, then σ(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, we can state: 10 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' rλ(G) ∈ O(2n/4) for a graph G of order n that is a union of C4’s, for the specific labelling scheme λ that we described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Towards optimal sum labellings We know that σ(kC4) ∈ {2, 3} (Lemma 2), and it is known that σ(C4) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Can we possibly also show that σ(2C4) = 2 or even σ(3C4) = 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let us try a bit of algebra, assuming arithmetic progression labellings of the two considered C4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' x x + d x + 3d x + 2d 2x + d 2x + 3d 2x − d 2x + 5d 4x + 4d 4x + 8d Figure 4: An algebraic approach to the C4 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The idea of Figure 4 is to find one of the three isolates of the second cycle within the labels of the first cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The only way this could happen is for the isolate 4x = (2x+d)+(2x−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Clearly, 4x ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If 4x = x+d, then 3x = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This contradicts the label 2x − d, which implies that 2x > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If 4x = x + 2d, we conclude 3x = 2d, so that x is even and d is divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The smallest numbers satisfying these conditions are x = 2 and d = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' These divisibility conditions also enforce that all other labellings of this form have to be scalings of this minimal labelling by some constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, if 4x = x+3d, then x = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, the number 2x+d = x+2d would occur twice as a vertex label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, under the conditions that our first cycle is labeled as in Figure 4, Figure 5 basically shows the only possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Notice that this labelling contains the NTAP 2 − 5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 2 5 11 8 19 13 1 7 20 32 Figure 5: A minimal way to label a 2C4 with two isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Could scaling help to also label 3C4 with our strategy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The somewhat surprising answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' First, we look at a concrete example in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The trick consists in the following steps: 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Multiply all labels used so far by a sufficiently large constant z > 2, which is four in our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We actually need that (modulo z) z − 1 ̸= z + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To ease our inductive argument, let us always pick z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Pick the smallest three labels of the first cycle, which is x = 8, x+d = 20, x + 2d = 32 in our example, and select numbers a, b, c, e to label the third cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To avoid unwanted edges, choose a = x/2 + 1 (recall that x must be an even number), b = x/2 − 1, c = (x + 2d)/2 + 1, e = (x + 2d)/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Observe that the isolates of the 2C4-construction remain untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, since our labelling of 2C4 contains a NTAP, the proposed la- belling of 3C4 contains a NTAP too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 8 20 44 32 28 52 4 76 5 3 15 17 80 128 Figure 6: A way to label a 3C4 with two isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 32 80 176 128 112 208 16 304 20 12 60 68 17 15 63 65 320 512 Figure 7: A way to label a 4C4 with two isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As the 2C4-construction remains untouched up to scaling, we can ac- tually repeat this argument, which could give the labelling of a 4C4 as in Figure 7, and this type of argument continues to prove by induction on k: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' σ(kC4) = 2 for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, the corresponding labelling contains a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let us describe some details of the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For our inductive argument to work, we make the additional claim that the three smallest 12 numbers x, y, z labelling the first cycle form an arithmetic progression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', there is a number d such that y = x + d and z = x + 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, d is not a label of any vertex, so that the labelling satisfies NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The induction basis for k = 2 was given above and satisfies NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let us assume that the labelling strategy works for some specific k = K ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' When we multiply all labels of the first K cycles by four, then this will not change the fact that (exactly) the edges of the K cycles are described by these numbers, plus the two isolates that remain as isolates in the overall labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also some NTAP is found after the modification by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labelling of the (K + 1)st cycle builds upon the smallest three labels x, x + d, x + 2d of the first cycle, choosing a = x/2 + 1, b = x/2 − 1, as well as c = a + d and e = b + d as labels of the last cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As we multiplied all original numbers by four, x is an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, a + b = x, a + e = b + c = x + d and c + e = x + 2d, so that all wanted edge labels can be found as vertex labels on the first cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' By way of contrast, the unwanted edges corresponding to a + c = 2a + d = x + d + 2 and b + e = 2b + d = x + d − 2 cannot be found as vertex labels, because all vertex labels of the first K cycles (and also the isolates) are divisible by four, including the label x + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, as all ‘new labels’ are odd and all ‘old labels’ are even, an edge between an ‘old vertex’ and a ‘new vertex’ must be labelled with a ‘new label’, and this also implies that only the two bigger ‘new labels’ c and e could possibly serve as edge labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, as c is one congruent four, this must match the only other label that is one congruent four, which is a, as all ‘old labels’ are divisible by four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, the question is if c − a = d is an ‘old label’, which is clearly not the case by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Similarly, b and e are three congruent four, but e − b = d and the same argument applies in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ Notice that in the recursive labelling algorithm hidden in the previous proof, the assumption that d does not occur as a vertex label is crucial, as otherwise there would be an unwanted edge between a and d, because we have the vertex label a + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In contrast to the labelling strategy described in the previous subsec- tion, and in particular analyzed in Lemma 3, we obtain a worse relation concerning the growth of the range for this new labelling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There is a labelling strategy λ for disjoint unions of C4’s such that rλ(G) ∈ O(2n/2) for a graph of order n which is a collection of n/4 many C4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Although also the size of the smallest label grows in this magni- tude, it suffices to estimate the size of the largest label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For 2C4, this is 32 = 2 · 28/2, as described in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As this largest label is always multi- plied by 4 = 24/2, we get 2·2n/2 = 2·2(n−4)/2 ·24/2 by induction, considering an n-vertex graph which is a collection of n/4 many C4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ We can generalize the construction of Lemma 4 to get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G be a graph with δ(G) = σ(G) = 2 such that there is a sum-optimal labelling λ of the sum graph H = G + N2 such that λ(V (H)) contains a NTAP, then there is a sum-optimal labelling λ′ of the sum graph H′ = G + C4 + N2 such that λ(V (H′)) contains a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For example, consider C3 + C4, starting with a labelling 1 − 3 − 4 of the C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Now, the isolates are 5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Observe that 1, 3, 5 is an arithmetic progression whose offset 2 is not a vertex label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, we can multiply the numbers by 4 to get a labelling 4 − 12 − 16 of the C3, with isolate labels 20 and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, we label the C4 as 1 − 3 − 9 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Clearly, the same isolate labels of 20 and 28 suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proposition will come in handy when finally combining the results of this subsection with that of the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The importance of the arithmetic progression becomes also clear when revisiting Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Unfortunately, as already described in Lemma 5, the range will grow exponentially with base √ 2 if this proposition is applied repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Collections of cycles without 4-cycles We first discuss labelling strategies for collections of cycles without C4, but Proposition 2 immediately shows a way how to add C4 afterwards, as we explain in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Dealing with long cycles We will consider Fibonacci-labellings of cycles Cn, with n > 4,2 leaving the case of triangles to be treated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' λn(x, y) : (x, x + y, 2x + y, 3x + 2y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , F(n)x + F(n − 1)y) with isolates (F(n) + 1)x + F(n − 1)y and F(n + 1)x + F(n)y, using the Fibonacci sequence F : (1, 1, 2, 3, 5, 8, 13, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=') with F(0) = 0 for conve- nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 2Collections of such longer cycles were treated in [13] in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 14 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For any n ≥ 3, n > 4, and any 1 ≤ x ≤ y, λn(x, y) gives a sum labelling of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In fact, we can consider the labelling scheme λ(x, y)(k) = F(k)x+F(k−1)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Notice that the corresponding range function grows as ϕn for Cn, where ϕ is the golden ratio number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The parameters x and y (x < y) give us great flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We could start labelling the first of a certain number of longer cycles, starting with x1 = 1 and y1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If the first cycle has length n1, the last of its vertices would be labelled F(n1)x1 + F(n1 − 1)y = F(n1) + F(n1 − 1) = F(n1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The two isolates (F(n1)+1)+F(n1−1) = F(n1+1)+1, F(n1+1)+F(n1) = F(n1+2), giving us a new pair (x2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If the second cycle has length n2, the last of its vertices would be labelled F(n2)x2 + F(n2 − 1)y2 = F(n2)(F(n1 + 1) + 1) + F(n2 − 1)F(n1 + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We get similar expressions for the isolates, which will again form the start (x3, y3) of labelling the next cycle etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We explain this labeling strategy by an example, a collection of three C5 in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The disadvantage of this strategy comes from the fact that we do not see an arithmetic progression in it such that its offset is not a vertex label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In particular, how do we label C5 + C4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' However, there is a remedy to it: Singla, Tiwari & Tripathi [10] showed that (for rσ, the spum number) rσ(Cn) ∈ [2n − 2, 2n − 1] for n ≥ 4, and rσ(Cn) = 2n − 1 for n ≥ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Namely, for odd n ≥ 5, they propose the label set L(Cn) = [n − 3, 2n − 4] ∪ {3n − 6, 3n − 4} ordered as (n − 3, n − 1, 2n − 5, n + 1, 2n − 7, n + 3, 2n − 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , 2n − 4, n − 2, n − 3) For instance, this gives the labelling (2, 4, 5, 6, 3) of a C5, with isolates 9, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Unfortunately, this is not a valid labelling as claimed in the paper, as we get the unwanted edge between the vertices labelled 2 and 9, adding up to the label 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labelling works for n = 7, though, where we get (4, 6, 9, 8, 7, 10, 5) with isolates 15, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Namely, we find that the difference 3n−6−(3n−4) = 2 between the two isolate labels only occurs in [n−3, 2n−4] when n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In [10], another labelling was proposed for even n ≥ 4:3 L(Cn) = [n − 2, 2n − 3] ∪ {3n − 5, 3n − 3} ordered like (n − 2, 2n − 3, n, 2n − 5, n + 2, 2n − 7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , 2n − 4, n − 1, n − 2) 3Other than claimed, the proposed labelling actually does not work for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 15 Thus, for n = 6, the C6 can be labelled (4, 9, 6, 7, 8, 5), with isolates 13, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In all cases, we clearly find a NTAP, for instance for the proposed C6- labelling 4, 5, 6 with offset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As this is of importance for our algorithm, let us show that there does indeed exist a sum labelling of the C5 that satisfies the conditions we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labelling (1, 2, 7, 9, 3), with isolates 4, 12, 16 contains the arithmetic progression 2 − 7 − 12 whose offset 5 is not a label of any vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This looks worse than what the Fibonacci labelling would deliver, as we need three isolates, but it is in fact better, as 4, 12, 16 might be seen as the start of a Fibonacci labelling on the second cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proves that σ(C5 + Cn) = 2 if n ̸= 4, and accordingly an optimal labelling can be given that contains a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For instance, C5 + C3 can be labelled with {1, 2, 3, 4, 7, 9, 12, 16}, with isolates {20, 28}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Only for the next cycles, we employ the Fibonacci scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This preserves the property to have a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labelling of C5 + C4 with the C5 labelled as (2, 4, 6, 10, 16) and the C4 labelled as (1, 9, 18, 26) (with isolates 27, 44) contains the NTAP 10 − 27 − 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The C5-labelling follows a Fibonacci scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As 1 + 9 = 10, one C4 edge label is in the C5, while its other edge labels are in the isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ This labelling scheme can be generalized as follows: C5 : (a, b, a + b, a + 2b, 2a + 3b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' C4 : (c, 2b + c, 3a + 3b, 3a + 5b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' isolates : 3a + 5b + c, 6a + 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The above labelling requires that a = 2c (to ensure that a+2b = 2b+2c for the edge connecting 2b + c and c), and b ̸= 3c (to avoid the unwanted edge a + (a + b) = b + 4c being represented by 2b + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To conclude this section, we show how to label 3C5 in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Figure 8 shows a labelling were we strictly follow a Fibonacci scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If we use Lemma 7 at the beginning (with the advantage of showing an arithmetic progression as desired), we arrive at Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 16 1 2 3 5 8 9 13 22 35 57 66 92 158 250 408 Figure 8: How to label a collection of cycles, here three C5’s, with isolates 474, 658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 1 2 7 9 3 12 16 28 44 4 48 72 120 192 312 Figure 9: How to label a collection of cycles, here three C5’s, with isolates 360, 504, and NTAP 2 − 7 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Dealing with triangles We will prove the following assertion about collection of triangles (C3’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Any Fibonacci labelling scheme for a non-empty collection of triangles gives a valid sum labelling that contains a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More precisely, consider λ′ n(x, y) : (x, x+y, 2x+y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 3x+y, 3x+2y, 6x+3y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 9x+4y, 9x+5y, 18x+9y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=') as a labelling scheme serving for ℓ many C3’s, with isolates 3ℓx + ⌊3ℓ/2⌋y, 3ℓx + ⌈3ℓ/2⌉y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This scheme is different in terms of growth compared to the schemes set up before for longer cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Yet, it can be embedded in an inductive construc- tion of a labelling of any number of cycles excluding C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To finally co-ordinate such a labelling with subsequently labelling a col- lection of C4, we need to satisfy a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' First, observe that y (in λ′ n(x, y)) is not the label of any vertex if y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, in order to obtain an arith- metic progression, we might find x+2y = 3x+y, or y = 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In other words, we propose the labelling scheme λ(3) n (z) = λ′ n(z, 2z) : (z, 3z, 4z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5z, 7z, 12z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 17z, 19z, 36z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 53z, 55z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' ) 17 2 4 6 10 16 18 26 1 9 27 44 71 Figure 10: How to label C5 + C4 + C3, with isolates 98, 115, and NTAP 10 − 18 − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' for collection of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For example, for a single C3, we can take the labelling (1, 3, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5, 7), with z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If we consider the sequence g(n) of smallest labels per cycle in this sequence of labels, assuming z = 1, we get the recursion g(1) = 1 and g(k) = 3g(k − 1) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proves that g(k) ∈ Ω(3k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, the range of the labelling scheme λ(3) n (z) grows exponentially, similar to ( 3√ 3)n with the number n of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We can hence use these schemes to label any collection of cycles without C4, simply by interpreting the two isolates ι1 and ι2 necessary to label a collection of ℓ cycles (by induction hypothesis) as the first two vertices, say, x and x + y, of the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, it is clear that any collection of cycles that contains at least one triangle and that is labelled this way has a NTAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hence, we can apply Proposition 2 to add a collection of C4 on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' At each time, we only need two isolates for this collection of cycles, apart from one exception, when the whole collection only contains one C5, where a special labelling was described in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proves our main theorem for 2-regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Although our labelling strategy λ for 2-regular graphs G attains σ(G), one can see that rσ(λ) ∈ O(ϕn) in the worst case, where ϕ is the golden ratio number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' But if the cycle collection contains only smaller cycles, the growth rate becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Nonetheless, it stays exponential, and it is unclear if there are labelling strategies for 2-regular graphs whose range stays polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As a final comment, notice that the sequence of labellings that we pro- pose, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', our labelling strategy, is not the only possible one, as shown in Figure 10, where the labelling of C5 + C4 as shown in Lemma 8 is finally combined with labelling a C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Our standard labelling strategy would be a bit worse, as shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 18 4 8 28 36 12 5 3 25 23 16 48 64 Figure 11: Labelling C5 + C3 + C4, with isolates 124, 140, and NTAP 8 − 28 − 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Bringing paths into the game We are first discussing a general situation that we face after having dealt with all cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Here, we have to distinguish two cases: either, this cycle collection has sum number two, or it has sum number three, which means, it is a single C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Dealing with cycle collections of sum number two Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G = (V, E) be a graph with σ(G) = 2, testified by a labelling λ with isolate labels ι1 and ι2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If ι1 + ι2 ̸= λ(u) + λ(v) for any two vertices u, v ∈ V , then σ(G + Pk) = 1 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Assume ι1 < ι2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Recall the Fibonacci labelling scheme for paths (Equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We propose to use λφ ι1,ι2 to label Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Clearly, ι2 and ι1 + ι2 are the two biggest labels, labelling the second and third vertex on the path (or the isolate if k = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This is an invariant that is maintained by the Fibonacci scheme: the labels of the ℓth and (ℓ + 1)th vertex on the path are always greater than any previous labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' By this and due to the assumption that ι1+ι2 ̸= λ(u)+λ(v) for any two vertices u, v ∈ V , this labelling cannot introduce unwanted edges and is hence valid for G + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, it will leave us with one isolate only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As δ(G + P) = 1, the labelling is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ Notice that the argument also works if σ(G) > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' just pick the largest isolate label plus any other isolate label to produce the label λ3 (or ι if the added path has length two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This proves the following fact: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G = (V, E) be a graph with σ(G) ≥ 2, testified by a labelling λ with largest isolate labels ι1 and ι2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If ι1 + ι2 ̸= λ(u) + λ(v) for any two vertices u, v ∈ V , then σ(G + Pk) ≤ σ(G) − 1 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 19 Notice that this argument is different from the (more general) one pre- sented in [14] where σ(G1 + G2) ≤ σ(G1) + σ(G2) − 1 is proved under the assumption that optimum labellings λ1 of G1 and λ2 of G2 exist such that there is an element in λi(V (Gi)) that is relatively prime to the largest ele- ment of λ3−i(V (G3−i)) for i = 1 or i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Also, observe that the labels will grow exponentially by a Fibonacci labelling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Combining a 4-cycle with paths Given the ideas of Lemma 4 and the results so far, one might be tempted to think that the sum number of every graph G of maximum degree 2 with a disjoint copy of C4 is equal to the minimum degree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Our next result shows this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' σ(C4 + P2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Before proving Proposition 4, we require some observations about C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' These observations also provide an indication as to why C4 is different from all the other cycles (as far as the sum number is concerned, at least).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It was already shown by Harary [2] that σ(C4) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Here we present a reason for this in the following, as it indicates the way how lower bounds on σ can be shown when the minimum degree criterion (proving σ(Cr) ≥ δ(Cr) = 2 for each r ≥ 3) is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let C4 + G be a graph without isolates, and let H be a sum graph of C4 + G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, all vertices corresponding to edge sums of the C4 lie in H − C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In particular, this means that every sum labelling of a C4 is exclusive,4 and that hence σ(C4) = 3 holds because of the next lemma (Lemma 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof (Proof of Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We will prove this by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let the vertex labels of the C4 be (a, b, c, d) in cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Assume to contrary that a + b = c (due to the symmetry of C4, all other cases are similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, we claim that all of the following are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (i) There is a vertex labelled b in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (ii) There is a vertex labelled a + d in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 4This means that edge labels are among the isolate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 20 (iii) There is a vertex labelled a + b + d in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Since b lies in C4, (i) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Since (a, d) is an edge and H is a sum graph, (ii) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, since (c, d) is an edge, H is a sum graph, and c+d = a+b+d, (iii) is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Now, since H is a sum graph, (i), (ii), (iii) together imply that there must be an edge between the vertex labelled b and the vertex labelled a + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' But b has exactly two neighbours, labelled a and c, so a + d must be one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We will show that either case leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If a = a + d, then d = 0, which is impossible, as H is a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If c = a + d, then c = a + b implies that b = d, which is impossible, as H is a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let C4 + G be a graph without isolates, and let H be a sum graph of C4 + G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let S be the set of numbers that correspond to the four edge sums of the C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, |S| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We will show that |S| ≤ 2 leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' That is, the four edges of the C4 must have at least three distinct edge sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let the vertex labels of the C4 be (a, b, c, d) in cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Two edges that share a vertex cannot have the same edge sum, because then there would be two vertices with the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, the only way that the C4 can have only two distinct edge sums is if both the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' a + b = c + d a + d = c + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Subtracting the first equation from the second, we obtain that b−d = d−b, or b = d, which is impossible in a sum graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let C4 + G be a graph without isolates, and let H be a sum graph of C4+G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let S be the set of numbers that correspond to the four edge sums of the C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If |S| = 3, then the three numbers in S are in arithmetic progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let (a, b, c, d) be a labelling of the C4 such that a + b = c + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let sum = a + b = c + d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' diff = c − a = b − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 21 We will show that the labels of the three isolates are iso1 = sum − diff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' iso2 = sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' iso3 = sum + diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labels of the edges (a, b) and (c, d) are equal to iso2, due to the definition of sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As for the labels of the edges (a, d) and (b, c), we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' a + d = (c + d) − (c − a) = sum − diff = iso1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' b + c = (a + b) + (c − a) = sum + diff = iso3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Since iso1, iso2, iso3 are clearly in arithmetic progression, this completes the proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ Finally, we are ready to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof (Proof of Proposition 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Label the C4 as (1, 7, 13, 19), the P2 as (20, 32), and the two isolates as 8 and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is easy to check that this is a sum graph, and thus σ(C4 + P2) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' To prove that σ(C4 + P2) ≥ 2, assume to contrary that σ(C4 + P2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let the labels of the vertices of the P2 be (b1, b2) (assume b1 < b2), and the isolate be b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Recall that every C4 has at least three distinct edge labels (Lemma 11), and none of those labels can be present in the vertices of the C4 itself (Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, the only option is that the edge labels of the C4 are b1, b2, b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Furthermore, we know that whenever C4 has exactly three edge labels, those three numbers form an arithmetic progression (Lemma 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Now, observe that the largest label of P2 (namely, b2) must be the largest label of the graph G = C4 + P2, since G has only one isolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, for the edge label b1 + b2 of the P2, we have: b1 + b2 = b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (3) As mentioned in the previous paragraph, since b1 < b2 < b3 are the three edge labels of the C4, they are in arithmetic progression, implying that b3 − b2 = b2 − b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (4) With Equation (3) and Equation (4), we get b2 = 2b1 and b3 = 3b1, or bi = ib1 ∀ i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' (5) 22 Consider a P3 subgraph of the C4 such that one of the two edges of the P3 is labelled b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More precisely, let the P3 be (a1, a2, a3) such that a1 + a2 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Since a1 ̸= a3, the edge (a2, a3) cannot be labelled b2, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Thus, (a2, a3) is labelled either b1, or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' That is, a2 + a3 is equal to either b1, or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If a2 + a3 = b1, then a1 − a3 = (a1 + a2) − (a2 + a3) = b2 − b1 = 2b1 − b1 Using (5) = b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If a2 + a3 = b3, then a3 − a1 = (a2 + a3) − (a1 + a2) = b3 − b2 = 3b1 − 2b1 Using (5) = b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, either a1 = b1 +a3, or a3 = b1 +a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In other words, either (b1, a3) is an edge, or (b1, a1) is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In either case, there is an edge between the C4 and the P2, which is a contradiction because they are supposed to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ On the other side, we can prove: Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' σ(C4 + Pk) = 1 for all k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If k = 3, label the C4 as {1, 3, 9, 11} and the P3 as {12, 4, 16}, with the isolate being 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If k ≥ 4, then the first four labels of the path Pk = (v1, v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , vk) are λ(v1) = 12, λ(v2) = 4, λ(v3) = 16, λ(v4) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' After that, we simply continue in the Fibonacci fashion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', λ(vi+1) = λ(vi) + λ(vi−1), with the label of the isolate being λ(vk) + λ(vk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' It is easy to check that no unwanted edges are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ The general algebraic strategy can be best seen by labelling the C4 with {1, 3, 9, 11}, with a < b being the smallest numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We assume that a + d = b + c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', d = b + c − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Moreover, we label the three path vertices with {a+d, a+b, (a+d)+(a+b) = a+2b+c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In order to save on isolates, we also require that c+d = b+2c−a equals (a+2b+c)+(a+b) = 2a+3b+c, 23 which implies c = 3a + 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In summary, given small numbers a < b, we construct the further labels of C4 as 3a + 2b and 2a + 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, the labels on the path would be 3(a + b), a + b, 4(a + b), with the isolate 5(a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If we want to label C4 + Pk with k ≥ 4, it is possible to save on the size of the labels by starting with labelling the C4 with {1, 2, 6, 11} and the P4 with {17, 3, 8, 12}, plus one isolate labeled 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Further savings are possible if we label the C4 with {2, 5, 8, 11}, as done as a standard throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The first five labels of the path Pk, k ≥ 5, with vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' , vk are then: λ(v1) = 26, λ(v2) = 13, λ(v3) = 7, λ(v4) = 19, λ(v5) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' If k = 5, then 39 would be the label of the isolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Otherwise, we just continue in a Fibonacci-style, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=', λ(vi+1) = λ(vi) + λ(vi−1), with λ(vk) + λ(vk−1) being the isolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Again, no unwanted edges are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There is only one case left over to complete the picture: Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' σ(C4 + 2P2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' By Proposition 4, σ(C4 + P2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' The labelling satisfies the requirements of Proposition 3, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Combining cycles with more than one path The following proposition also covers the case of pure path collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let G be a graph with σ(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, σ(G + Pk) = 1 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let ι be the label of the isolate of a labelling λ of G certifying its sum number to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, ι is bigger than any vertex label of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' There- fore, labelling Pk with the Fibonacci labelling scheme λφ ι,2ι, as introduced in Equation (2) in general form, labels G+Pk (together with λ) with only one isolate, not creating conflicts, as all edge labels of G are smaller than 2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ We already saw (or will see soon) that a cycle collection plus one path has sum number one with one exception, which is C4+P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As we will fix the only remaining case of C4 + 2P2 separately in Lemma 14, we can conclude: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Let C be a collection of cycles and P be a collection of at least two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Then, σ(C + P) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Except for the case of C4+P2, we know that σ(C +Pk) = 1 for any collection of cycles C and any k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' As we consider paths in decreasing length, we will pick a cycle of length at least three to be considered first if there is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Therefore, if P is a collection of at least two paths, then we can conclude σ(C + P) = 1 either by Proposition 5 directly, or by first taking Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' □ 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Conclusion and Open Problems We have explained that the labelling of a C5 as proposed in [10] is not working correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This leaves the spum-minimization problem open for this particular small graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' But this question easily generalizes to nearly all graphs with maximum degree of at most two as discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' In most cases, we only found labellings with labels of exponential size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' This might be necessary, but for such statements, we do not have any proof idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Our main result concerns the sum number of (all) graphs of maximum degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Kratochv´ıl, Miller and Nguyen posed in [3] two conjectures that are tightly related to our paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' we will formulate them as questions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Given two graphs G1, G2 with σ(G1) = σ(G2) = 1, is it true that the sum number of their disjoint union is always one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' More generally: given two graphs G1, G2, is it true that σ(G1 +G2) ≤ σ(G1) + σ(G2) − 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Observe that we did resolve the first question if G2 is a path (Proposition 5), but the general question is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Upon some thought, it can be seen that the general question is related to the following natural combinatorial question: Find a characterization of all graphs with sum number one, also known as unit graphs in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' We also refer to a recent paper [15] that studies variations of this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Finally, a slightly weaker but more structured notion of sum labelling (called arithmetic graphs) could lend some ideas that might help in resolving this question (and more optimisti- cally, the second question) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Apart from these combinatorial questions, the basic complexity ques- tions concerning the graph parameter σ and rσ are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' For instance, is it NP-hard to decide if, given a graph G and a number k, σ(G) ≤ k holds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' One of our own motivations to study graphs of maximum degree two was to see if one could use the operation of graph union to piece gadgets to- gether for this and similar questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' But we are still far from this, as even these seemingly easy questions concerning graphs of maximum degree two are non-trivial to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' References [1] J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' Hegde, Arithmetic graphs, JGTh 14 (3) (1989) 275–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9A0T4oBgHgl3EQfPv8L/content/2301.02178v1.pdf'} diff --git a/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/2301.13444v1.pdf.txt b/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/2301.13444v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2a23bc6caf60cba6d3895a2c3ae5ae0da0c2e4f --- /dev/null +++ b/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/2301.13444v1.pdf.txt @@ -0,0 +1,4751 @@ +Rethinking Soft Label in +Label Distribution Learning Perspective +Seungbum Hong, Jihun Yoon, Bogyu Park, and Min-Kook Choi∗ +AI Dev. Group +Hutom +Seoul, Republic of Korea +mkchoi@hutom.io +Abstract +The primary goal of training in early convolutional neural networks (CNN) is +the higher generalization performance of the model. However, as the expected +calibration error (ECE), which quantifies the explanatory power of model inference, +was recently introduced, research on training models that can be explained is in +progress. We hypothesized that a gap in supervision criteria during training and in- +ference leads to overconfidence, and investigated that performing label distribution +learning (LDL) would enhance the model calibration in CNN training. To verify +this assumption, we used a simple LDL setting with recent data augmentation +techniques. Based on a series of experiments, the following results are obtained: 1) +State-of-the-art KD methods significantly impede model calibration. 2) Training +using LDL with recent data augmentation can have excellent effects on model cali- +bration and even in generalization performance. 3) Online LDL brings additional +improvements in model calibration and accuracy with long training, especially +in large-size models. Using the proposed approach, we simultaneously achieved +a lower ECE and higher generalization performance for the image classification +datasets CIFAR10, 100, STL10, and ImageNet. We performed several visualiza- +tions and analyses and witnessed several interesting behaviors in CNN training +with the LDL. +1 +Introduction +The supervision of a convolutional neural network (CNN) using a hard label has been very successful +in most image classification problems [1, 2, 3]. However, in the training of a CNN using a hard label, +as the number of weights of the network increases, [4] analyzed the overconfidence of the network +prediction. To handle this phenomenon, [4] proposed the expectation of calibration error (ECE) to +estimate the confidence of the model, and several approaches for calibrating the overconfidence of +deep learning models were suggested, but they were not correlated with generalization performance +and model calibration. Recently, several studies have introduced that data augmentation is effective +for model generalization as well as calibration [5, 6], but the results are not significant in terms of +generalization performance. +Label distribution learning (LDL) is designed for effective training through label distribution when +the types of labels for supervision are difficult to define discretely and approaches the label generation +(or enhancement) process as an optimization problem [20, 21]. Typically, LDL has been applied to +applications that include inherent label ambiguity, such as facial age estimation, head pose estimation, +facial emotion estimation, multi-label learning, partial multi-label learning, and video summarization +[21, 22]. From a LDL perspective, label smoothing [14, 17, 19] is considered a subset of LDL.We +∗Corresponding author. +Preprint. Under review. +arXiv:2301.13444v1 [cs.CV] 31 Jan 2023 + +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +cat +dog +person +cat +dog +person +cat +dog +person +cat +dog +person +person +Figure 1: The difference between the supervision using the hard label (blue box) and using the +soft label (red box) for image classification. An illustration of the LDL for the cat class at the left is +shown, and the reliability diagrams at the right show comparisons of the traditional hard label-based +and LDL-based classification accuracy and ECE. Our LDL-based training successfully achieved +better classification accuracy and lower ECE simultaneously. +are inspired by the basic concepts of LDL and assume that the LDL potentially overcomes the +discrepancy between the one-hot label-based training and the maximum confidence- based testing. +To introduce this idea in a simple way, we exploited soft labels from teacher networks as a baseline +distributed label. To learn the label distribution online differ from the former optimization approach, +recent data augmentation techniques that merge labels and data during training were simultaneously +applied. +By applying the on/offline label distribution learning scenarios, we simultaneously obtained an +improvement in the model generalization and calibration without additional regularization or archi- +tecture modification. The left section of Figure 1 shows examples of the difference between hard +label and LDL–based supervision for cat recognition. The graphs at the right of Figure 1 show the +reliability diagram [4] when different settings of modern KD approaches in the same CNN model are +applied for CIFAR100. To verify the strength of LDL for model generalization and calibration, we +performed a series of image classification tasks on datasets such as CIFAR10, 100 [23], STL10 [24], +and ImageNet [25]. Based on a series of experiments with image classification, we confirmed that +most recent KDs cause severe overconfidence, which impedes model calibration, and even a simple +LDL approach can achieve better classification accuracy and suppression of model overconfidence. +2 +Related Works +Model calibration. ECE is an error that measures whether the prediction of the neural network +can accurately estimate the true likelihood of the input data of the trained classes [4, 40]. In [4], +temperature scaling was proposed in a way that can effectively be corrected and the reliability diagram +is used to visualize model confidence for CNNs. In [30], various structural dropout methods and +experiments on the drop rates according to each method were applied to the CNN model to analyze +the correlation between model accuracy and ECE. VWCI [31] reduced the ECE and improved the +recognition performance by defining a confidence integration loss as a probabilistic regularization +term defined from a Bayesian model using multiple inferences based on probabilistic depth and +dropout. It has also been reported that the AvUC loss based on uncertainty estimation in the model +also aids in model calibration [32]. In addition to this, model training with mixup augmentation +has been demonstrated to be effective in model calibration. However, it did not achieve much in +improving the generalization performance of the model in preparation for the correction effect [5, 6]. +Label smoothing and label distribution learning. Label smoothing was proposed to soften the hard +label in the training process according to the given coefficient to prevent overconfidence and improve +generalization performance [17, 18]. In [18], the authors analyzed the effect of label smoothing on +deep neural network training by visualizing the penultimate fully connected layer of deep neural +networks. According to the analysis results, there is evidence that the trained teacher network applied +with label smoothing in the KD scenario can invalidate the effect of student model training. In +recent studies [19, 28], the effect of label smoothing on teacher networks in the KD scenario was +analyzed in more detail to extend the research results [18]. In [19], a quantification method that +label smoothing erases meaningful information in the teacher network logit was proposed. In [28], +the relationship between KD and label smoothing from the bias-variation perspective was analyzed. +2 + +ReliabilityDiagram(T:R110,S:R56) +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +Vanilla(72.19) +TS(72.19) +0.2 +LS(72.56) +KD(72.63) +CRD+KD(75.7) +WSL(74.99) +Ourbest(75.85) +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceReliabilityDiagram(T:R200,S:R18) +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +Vanilla(77.28) +TS(77.28) +0.2 +LS(78.89) +KD(77.57) +CRD+KD(80.27) +WSL(79.94) +Ourbest(81.15) +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceOFrom the LDL perspective, Label smoothing can be regarded as a possible solution for LDL through +constant softening of the hard label. We included label smoothing in our comparisons as one of the +baselines to suppress overconfidence [18]. +Knowledge distillation. Since the introduction of the KD [7], a vast number of approaches for +knowledge distillation have been proposed [8-16, 26, 27]. FitNet [8] proposed a KD method that +makes the feature maps of teacher networks similar. In recent years, various approaches have been +proposed from the perspective of representation learning, such as RKD [10], which achieved transfer +learning through geometric relations to the output of the model, CRD using metric learning [12], +mutual learning based [13], self-supervised learning–based KD [15], and weighted soft label-based +KD [16]. Among the variations of KD, the born-again network [9], which achieves transfer learning +through repetitive training of student models without the use of a teacher network, and similar to [9], +variations of KD that are free from teachers [14, 29] were also introduced. [18] and [19] explain the +relationship between KD and label smoothing of teacher networks through empirical experiments. We +argue that KD should be described in terms of LDL rather than label smoothing. We have observed +that modern KDs spoil model calibration to improve generalization performance. +3 +On/Offline Label Distribution Learning (LDL) +In this section, we briefly introduce the notations and approaches for KD and label smoothing (Section +3.1), which are basic prerequisites for our LDL-based approaches. Subsequently, on and offline +approaches for LDL are described in Sections 3.2 and 3.3. +3.1 +Preliminaries +Knowledge distillation. When the weight w of the last fully connected layer for the ith feature +input x and the output with the softmax function for the kth class is given as pk +i = +e(xi)T wk +�L +l=1 e(xi)T wl , the +softening output for the neural network is given by [7]. +¯pk +i (x) = +e((xi)T wk)/τ +�L +l=1 e((xi)T wl)/τ , +(1) +where τ is a temperature scaling parameter that determines size of softening, and the total loss +function L = (1 − λ)LCE(y, pθs) + λLCE(¯pθt, ¯pθs) for the teacher model θt and the student model +θs, where y is one-hot label and LCE(p, y) = − �K +k=1 yk log(pk). +Label smoothing The label smoothing for the hard label yi for the same feature input is given as +follows: +˜yk +i = (1 − α)yk +i + +α +K − 1, +(2) +where α is the smoothing coefficient, and the probabilities for each class except the hard target +corresponding to the kth class are evenly distributed as α/(K − 1). +Expected calibration error. The ECE for estimating the confidence of the neural networks proposed +in [4] is estimated for Ntest samples when the softmax output pi inferred for all test data and the +index with the maximum probability in the output is ˆci = argmax +i +(pi = k). +ECE = +M +� +m=1 +|Hm| +Ntest +� +1 +|Hm| +� +i∈Hm +1(ˆci = ci) − pi +� +, +(3) +where Hm is the index set and generates M interval bins of ((m − 1)/M, m/M] for Ntest samples. +Typically ECE is measured by the histogram for a bin of 0.1 size by setting M = 10. +Criterion of LDL with cross entropy loss. The main objective of cross entropy loss with LDL +perspective is given by: +3 + +( ) +To Be +Learned +hard label +output +① +Vanilla +To Be +Learned +soft label +output +Smarter +Model +KD +To Be +Learned +output +soft label +Smarter +Model +③ +Offline LDL +To Be +Learned +output +soft label +Smarter +Model 1 +Smarter +Model k +... +④ +Offline-En- +LDL +To Be +Learned +output +soft label +Smarter +Model 2 +Smarter +Model 1 +Smarter +Model k +... +output +soft label +output +soft label +⑤ +Offline-MT- +LDL +output +Smarter +Model +To Be +Learned +⑦ +output +⑥ +To Be +Learned +Label +smoothing +output +LS +soft label +② +⑧ +Online LDL +𝑥! +𝑥" +"𝑥 +⑨ +To Be +Learned +output +soft label +Online-En-LDL +𝑥! +𝑥" +"𝑥 +Smarter +Model 1 +Smarter +Model k +... +To Be +Learned +output +soft label +Online-MT-LDL +𝑥! +𝑥" +"𝑥 +Smarter +Model 1 +Smarter +Model k +... +output +soft label +hard label +Figure 2: Schematic of the training configurations. We used abbreviations to simplify the notation: +#1 learning from scratch (Vanilla), #2 label smoothing (LS), #3 knowledge distillation (KD), #4 soft +label (Off-LDL), #5 teacher ensemble for soft label (Off-En-LDL), #6 linear combination with the +multiple soft label (Off-MT-LLD), #7 soft label with data augmentation (On-LDL), #8 soft label +using data augmentation with teacher ensemble (On-En-LDL), #9 linear combination of multiple soft +label using data augmentation (On-MT-LDL). The red box represents the existing training method, +the green box represents the offline approaches, and the blue box represents the online approaches. +LCE(p, z) = − +K +� +k=1 +zk log(pk), +(4) +where z is a label vector that satisfies � +j zj = 1. Label smoothing or soft label by output of teacher +networks can be regarded as a specific solution for z (z = ˜y(α) or z = ¯pk +θt). We reformulated +the problem argmaxθ(E[hDtrain(x, y; θ)] − E[hDtest(x, y; θ)]) to find the CNN with the maximum +generalization performance in a specific image classification dataset D ∋ {Dtrain, Dtest} as follows: +argmax +θ,Z +E[hDtrain(x, z; θ)] − E[hDtest(x, y; θ)], +(5) +where ZDtrain ∋ {z1, ..., zNtrain} is a set of new labels for all training data. Equation (5) can be +solved as an optimization problem of finding a pair of the optimal labels for each input data (xi, zi) +such as the previously proposed LDL approaches [20, 21]. Since traditional approaches cannot +update z and θ simultaneously during deep neural network training, we applied simple but effective +on/offline approaches. For the simplicity, we used basic KD setting, which teacher network generate +new label set Z for target (student) neural network training. The Offline setting is a way to generate +Z as a soft label as the output of the teacher network. Z is not updated during training, but some +variations are possible depending on the way the ensemble of teacher networks. The online setting is +a way of continuously transforming Z while updating θ. Since it is difficult to presume an optimal Z, +we generated diverse labels with modern data augmentation techniques. As with offline settings, there +are several variations depending on the way the ensemble of teacher networks and label generation +process. Figure 2 shows different types of training configurations for baseline and LDL. +3.2 +Offline LDL +We simplified the problem by using the KD setting by the teacher output as the new label to extract +feasible solutions for each sample pair (xi, zi; θ). The offline LDL is illustrated in the green box in +Figure 2, such that the set of sample pairs (XDtrain, ZDtrain) under XDtrain ∋ {x1, ..., xNtrain} is +fixed during the training process. The cross-entropy loss for training with the label generated by the +teacher θt and the student model to be trained is θs is as follows: +LCE(p, ¯z) = − +K +� +k=1 +¯zk +i log(pk +i,θs), +(6) +where ¯z is defined by way of generating new labels. We used simple variations of ¯z = f(·) for +offline LDL (see Figure 2): soft label f(xk +i ; θt) (Off-LDL, #4), soft label with teacher ensemble +¯z = 1 +N +�N +n=1 f(xk +i ; θt,n), where N is the number of teacher models (Off-En-LDL, #5), and linear +combination of soft labels from multiple teachers − �N +n=1 +�K +k=1 ¯zk +i,θt,n log(pk +i,θs) (Off-MT-LDL, +#6). +4 + +3.3 +Online LDL +To reflect the objective of Equation (5) during training, it is necessary to update Z and θ simultane- +ously. We applied recent data augmentation techniques that can simply adopt online LDL. Among the +proposed data augmentation techniques, some techniques manipulate input data and label together. A +generalized form of augmentation technique considering data and labels together is +ˆx = M1 ⊗ x1 + ... + MP ⊗ xP +ˆy = λ1y1 + ... + λP yP , +(7) +where ˆx is an augmented sample of mixed label ˆy with �P λi = 1 up to P samples. M is a blending +mask equal to the data width and height, and satisfies �P Mi(u, v) = 1, where u and v indicate the +pixel location. Each augmentation algorithm is designed to stochastically determine the location and +size of each sample for blending and mainly follows a uniform distribution. M is provided differently +for each augmentation technique. Typically, When P = 2, x1 is defined as the target image, and x2 +is defined as the 0 image for CutOut [3]. We exploited mixup [33], CutMix [34], and RICAP [35] for +data augmentation, and online LDL with data augmentation was as follows: +LCE(ˆp, ˆz) = − +K +� +k=1 +ˆzk +i log(ˆpk +i,θs), +(8) +where ˆpk +i = +e(ˆxi)T wk +�L +l=1 e(ˆxi)T wl . Data augmentation applies equally to teacher models for label en- +hancement ˆzk +i , but the mixed label ˆy is not used for training. Similar to the offline approaches +corresponding to #5 and #6 in Figure 2, online LDL can be easily extended. The enhanced label +through the augmentation-based teacher ensemble is obtained as ˆz = 1 +N +�N +n=1 f(ˆxk +i ; θt,n) (On-En- +LDL, #8), and the linear combination of the augmentation-based soft labels from multiple teachers is +given as − �N +n=1 +�K +k=1 ˆzk +i,θt,n log(ˆpk +i,θs) (On-MT-LDL, #9). +4 +Experimental Results +Experimental setting. We performed a series of experiments on the CIFAR10, 100 [23], and STL10 +[24] datasets to verify the performance of the on/offline LDL. First, the major experiments were +performed with the teacher-student network configurations of the well-used ResNet architectures +[3]. We divided ResNet into small and large networks according to model size. Small size models +include ResNet20 (0.27M), 56 (0.85M), and 110 (1.7M) and large size models include ResNet18 +(11.18M), 50 (23.51M), and 200 (62.62M). For CIFAR10, 100, and STL10, a total of 240 epochs +was trained to start with an initial learning rate of 0.05, and 0.1 learning rate scaling was applied at +150, 180, and 210 epochs. The weight decay was set to 5.0 × 10−4, the batch size was set to 64. We +also tested a long training scenario with a basic training configuration, with the assumption that LDL +can fundamentally achieve better performance when the number of training pairs of sample and label +is large especially in online LDL. For long training, a total of 350 epochs was trained to start with +an initial learning rate of 0.05, and 0.1 learning rate scaling was applied at 150, 200, 250, and 300 +epochs2. In all ensemble (En) and multiple teacher (MT) settings, ResNet20, 32, 44, 56, and 110 +used together, were trained by the same learning scheduler. We measured the image classification +accuracy and ECE [4] to evaluate the performance. Visualization of reliability diagrams is provided +to intuitively check the strength of model calibration of the network in the same way as in [4]. +LDL with data augmentation. The augmentation algorithms applied for On-LDL are mixup [33], +CutMix [34], and RICAP [35], and the default hyper-parameter for data augmentation refers to the +original implementation of each algorithm. Table 1 shows the recognition results of the on/offline +LDL according to each data augmentation technique. All three algorithms showed rather poor +performance in small networks and were able to achieve significant performance improvement in +large networks. For long training, we only report the LDL of the augmentation technique with the +best performance. All training was performed on 3 different random seeds. We omit variations in +2The long training is marked with ’+’. +5 + +Table 1: Classification accuracy (%) and ECE of vanilla and each LDL setup for CIFAR100 depending +on each data augmentation methods [33, 34, 35]. +Teacher +ResNet110 +ResNet110 +ResNet200 +ResNet200 +Student (# param) +ResNet20 (0.27M) +ResNet56 (0.85M) +ResNet18 (11.18M) +ResNet50 (23.51M) +Vanilla +69.32±0.27/0.070 +72.28±0.09/0.123 +77.83±0.55/0.080 +78.80±0.17/0.107 +Vanilla [33] +67.29±0.17/0.127 +73.10±0.11/0.118 +78.71±0.29/0.131 +79.37±0.51/0.059 +Vanilla [34] +67.23±0.28/0.075 +73.99±0.16/0.066 +80.32±0.18/0.045 +81.73±0.07/0.038 +Vanilla [35] +68.46±0.12/0.070 +73.95±0.17/0.027 +80.10±0.05/0.039 +81.47±0.34/0.039 +Off-LDL +69.9±0.19/0.051 +73.59±0.36/0.085 +78.67±0.14/0.060 +79.19±0.37/0.087 +On-LDL [33] +68.42±0.12/0.130 +73.73±0.67/0.115 +79.76±0.31/0.121 +81.03±0.01/0.051 +On-LDL [34] +68.25±0.06/0.073 +74.44±0.16/0.060 +81.26±0.11/0.043 +83.09±0.05/0.030 +On-LDL [35] +68.90±0.05/0.063 +74.87±0.13/0.025 +80.57±0.04/0.056 +81.64±0.06/0.039 +On-LDL+ +69.41±0.17/0.073 +75.56±0.28/0.022 +81.76±0.25/0.034 +83.57±0.05/0.038 +Table 2: Classification accuracy and ECE for small size ResNets for CIFAR10 and STL10. +CIFAR10 +STL10 +Model +Model +Method +ResNet20 +ResNet56 +ResNet20 +ResNet56 +Vanilla +92.56±0.10/0.033 +93.88±0.08/0.038 +83.44±0.10/0.067 +84.15±0.31/0.074 +Label smoothing +92.41±0.25/0.052 +93.72±0.19/0.063 +83.54±0.22/0.111 +84.35±0.01/0.091 +KD (α=0.1,T=3) [7] +92.56±0.02/0.032 +93.94±0.03/0.038 +83.95±0.56/0.070 +84.62±0.41/0.070 +Off-LDL +92.62±0.18/0.028 +93.96±0.16/0.031 +84.14±0.42/0.055 +84.57±0.13/0.054 +Off-En-LDL +92.60±0.17/0.031 +94.07±0.01/0.032 +83.40±0.01/0.067 +84.03±1.07/0.069 +Off-MT-LDL +93.05±0.28/0.022 +94.15±0.17/0.022 +84.16±0.23/0.054 +84.52±0.09/0.060 +On-LDL +92.92±0.08/0.009 +94.04±0.09/0.013 +85.97±0.12/0.023 +86.24±0.37/0.029 +On-LDL+ +93.33±0.02/0.008 +94.32±0.05/0.011 +85.98±0.04/0.023 +86.61±0.15/0.029 +On-En-LDL +93.25±0.06/0.016 +94.48±0.09/0.016 +86.45±0.17/0.030 +86.92±0.42/0.034 +On-MT-LDL +93.14±0.05/0.006 +94.44±0.07/0.007 +86.44±0.14/0.008 +87.08±0.06/0.009 +On-En-LDL+ +93.71±0.02/0.017 +94.46±0.27/0.017 +86.82±0.03/0.030 +87.13±0.08/0.037 +On-MT-LDL+ +94.03±0.08/0.006 +94.41±0.02/0.006 +86.93±0.02/0.008 +86.92±0.08/0.010 +calibration scores, which are no significant differences. We selected the CutMix [34] or RICAP [35] +as a data augmentation for remain LDL experiments. Similar to the results of [5], with the mixup +augmentation, the improvement in accuracy was not significant, but a model calibration effect could +be seen in some cases. Rather, CutMix and RICAP achieved steady performance improvement and +better model calibration. Offline LDL achieved the best accuracy and model calibration performance +in ResNet20, and online LDL improved significantly as the size of the network increased. With the +ResNet50, an accuracy improvement of up to 1.9% and better model calibration was achieved than in +the case of data augmentation only. +CIFAR10 and STL10. Table 2 shows the evaluation results of the CIFAR10 and STL10 datasets +for LDL training. In the two relatively small datasets, we did not evaluate the large network, as +only the small network could provide sufficient performance improvement. In both CIFAR10 and +SLT10, online LDL utilizing multiple teacher networks showed improvements in accuracy and model +calibration. In CIFAR10, the accuracy increase of up to 1.6% and the ECE reduction effect of up +to about 80% were obtained in ResNet56 compared to the vanilla model. In STL10, the accuracy +increase of 3.39% and ECE improvement effect of about 85% were obtained in ResNet20 compared +to the vanilla model. In CIFAR10 and STL10, RICAP achieved better performance than CutMix. +CIFAR100. Table 3 shows the evaluation results of LDL in CIFAR100. In the upper part of Table 3, +the improvement in generalization performance is relatively insignificant for small networks. We +hypothesized that a large number of weights is required to sufficiently train the LDL-based label +variants and evaluated the small and large networks simultaneously on the CIFAR100 dataset. When +the model has a large number of weights, it is well trained on the label variation of the new label +distribution, and it is possible to achieve sufficient performance improvement and model calibration +without the multiple teachers. We obtained two main observations from the experiments with three +benchmarks: 1) The classification accuracy is mainly determined by the number of weights with LDL +approaches, and the influence of the number of parameters in the teacher model is not noticeable. +2) The effectiveness of model calibration steadily improves regardless of the number of parameters +in each model. We also compared the proposed LDL approaches with the SOTA KD methods +[12, 15, 16] on the CIFAR100 dataset. Table 3 shows the comparison results with small and large +student networks in terms of classification accuracy and ECE. In most cases, LDL-based methods +achieved a higher generalization performance and lower ECE simultaneously. Figure 3 shows this +phenomenon more dramatically: as the number of parameters in the student network is small, the +SOTA KD methods show a very high ECE score, have no explanatory power for model confidence, +6 + +Table 3: Classification accuracy and ECE of the LDL and the SOTA KDs for the CIFAR 100. Among +the LDL techniques, the performance of the models that achieved the highest accuracy or the lowest +ECE were recorded. +Teacher +ResNet110 +Student +ResNet20 +ResNet56 +ResNet110 +Vanilla +69.32±0.27/0.070 +72.28±0.09/0.123 +73.88±0.15/0.131 +Label smoothing +69.43±0.20/0.053 +72.87±0.17/0.020 +73.90±0.16/0.051 +KD (α=0.1,T=3) [7] +69.07±0.23/0.071 +72.76±0.11/0.118 +73.60±0.16/0.135 +CRD [12] +71.05±0.12/0.060 +74.82±0.11/0.107 +76.04±0.04/0.121 +CRD+KD [12] +71.29±0.23/0.129 +75.38±0.27/0.127 +76.67±0.46/0.125 +SSKD [15] +71.00±0.04/0.122 +74.88±0.14/0.119 +75.73±0.17/0.118 +WSL [16] +71.27±0.26/0.132 +75.08±0.19/0.133 +76.00±0.52/0.130 +Off-MT-LDL +70.75±0.15/0.019 +74.84±0.04/0.023 +76.35±0.18/0.016 +On-LDL+ +69.41±0.17/0.073 +75.54±0.24/0.029 +77.28±0.20/0.025 +Teacher +ResNet200 +Student +ResNet18 +ResNet50 +ResNet200 +Vanilla +77.83±0.55/0.080 +78.57±0.35/0.107 +79.47±0.58/0.101 +Label smoothing +78.95±0.06/0.085 +78.90±0.24/0.042 +79.59±0.42/0.037 +KD (α=0.1,T=3) [7] +77.73±0.16/0.077 +79.12±0.45/0.103 +80.10±0.23/0.100 +CRD+KD [12] +80.41±0.14/0.108 +80.34±0.07/0.118 +81.58±0.20/0.110 +WSL [16] +79.91±0.03/0.111 +80.25±0.12/0.114 +76.00±0.52/0.130 +On-LDL +81.26±0.11/0.043 +83.09±0.05/0.030 +83.88±0.28/0.038 +On-LDL+ +81.76±0.25/0.034 +83.57±0.05/0.038 +84.44±0.13/0.039 +Figure 3: Performance change by methods according to small and large student models for +CIFAR100. The left graph shows the accuracy difference with the vanilla training by the number of +weights of the student model, and the right graph shows the ECE difference with the vanilla training. +The LDL technique shows continuous improvement in model correction as the accuracy increases, +and the effect increases as the model size increases. On the other hand, SOTA KD methods have a +steady improvement in accuracy, but rather impede model reliability. +and result in overconfidence. LDL techniques were able to achieve better model calibration compared +to SOTA KDs as the number of parameters in the student network increased. Classification accuracy +showed up to 2.86% improvement in ResNet200 compared to the best performing CRD+KD, and an +improvement of at least 60% up to 88% compared to the SOTA KD methods in model calibration3. +ImageNet. We evaluated the ImageNet dataset [25] to verify LDL methods on the large-scale dataset. +For the evaluation of the ImageNet, we set up the multiple teacher and student network configurations. +The learning scheduler applied the configuration of ?, and the on/offline LDL methods were tested. +As shown in Table 4, similar to the other datasets, the similar tendency of improved accuracy and +model calibration was achieved simultaneously compared to vanilla and label smoothing. +Other model architectures. To validate the effect of LDL in various architectures, we performed an +evaluation according to ResNet200 teacher and ResNeXt [36], DenseNet [37], and DLA [38] student +networks in the CIFAR100. Table 5 lists the accuracy and model calibration improvements by LDL +approaches. Compared to the vanilla model in other types of architectures, there was an accuracy +improvement of about 3-5% and an ECE reduction of 20-66% was achieved. +3Types of teacher networks, long training results of SOTA KD methods, loss curves, and additional experi- +mental results and analysis are in the supplementary material. +7 + +LDL +(ours) + CRD+KD +[12] +WSL +[16] +4 +3 +2 +1 +0 +ResNet20 +ResNet56 +ResNet110LDL +(ours) - CRD+KD +［12] +WSL[16] +0.15 +0.10 +0.05 +0.00 +-0.05 +-0.10 +ResNet20 +ResNet56 +ResNet110LDL +(ours) +■ CRD+KD +[12] +WSL +[16] +5 +4 +3 +2 +1 +0 +ResNet18 +ResNet50 +ResNet200LDL (ours) + CRD+KD +[12] +WSL +[16] +0.08 +0.06 +0.04 +0.02 +0.00 +-0.02 +-0.04 +ResNet18 +ResNet50 +ResNet200Table 4: Classification accuracy (top1 / top5) and ECE for ImageNet dataset depending on each label +enhancement setup. +Teacher +ResNet152 +ResNet152 +ResNet152 +ResNet152 +Student +ResNet18 +ResNet50 +ResNet101 +ResNet152 +Vanilla +70.39/89.54, 0.014 +75.96/92.81, 0.032 +77.43/93.72, 0.045 +78.28/94.14, 0.050 +Label smoothing +70.40/89.52, 0.102 +76.56/93.12, 0.070 +78.36/94.05, 0.058 +78.82/94.30, 0.036 +Off-LDL +71.63/90.46, 0.013 +77.27/93.61, 0.021 +78.72/94.30, 0.024 +79.16/94.55, 0.021 +On-LDL +69.03/88.91, 0.078 +75.84/92.99, 0.017 +78.97/94.27, 0.016 +79.51/94.72, 0.022 +On-LDL+ +69.43/89.20, 0.062 +77.30/93.54, 0.023 +78.82/94.28, 0.022 +79.25/94.49, 0.021 +Table 5: Evaluation of different types of architectures. +Method +Student (# param) +ResNext29_4x64d (27.1M) [36] +DenseNet121 (6.95M) [37] +DLA (16.29M) [38] +Vanilla +80.30±0.14/0.050 +79.67±0.26/0.081 +77.27±0.50/0.099 +Label smoothing +80.73±0.37/0.151 +79.81±0.28/0.044 +79.07±0.33/0.039 +KD (α=0.1,T=3) [7] +80.38±0.34/0.052 +79.53±0.10/0.080 +77.67±0.21/0.101 +Off-LDL +80.59±0.10/0.039 +80.29±0.20/0.062 +78.12±0.06/0.081 +On-LDL +84.16±0.13/0.059 +83.35±0.21/0.028 +82.93±0.08/0.033 +ResNet18 +ResNet50 +Figure 4: Plotting the relationship between the F1 score and the average output confidence for +the GT class of the model according to the training methodology. The average output confidence +for each of the 100 classes of CIFAR100 and different shapes are marked for each F1 score. Values +in parentheses are the accuracy and ECE of each method. +Why does LDL work? Based on the evaluation results of four datasets, LDL is observed to have +an excellent effect on generalization performance and model calibration for CNN training. Figure 4 +is the result of the test set plotting the average confidence of ResNet18 and 50 for the ground truth +class on the x-axis and the F1 score for each class on the y-axis. In the case of one-hot label-based +training or offline LDL using only soft labels of a teacher trained on the one-hot label, most have an +average confidence score of 0.9 or higher regardless of the F1 score. Applying data augmentation or +label smoothing alleviates this over-confidence, but in the case of label smoothing, the confidence +distribution has an excessively large variance as a result of the forced softening. The case of online +LDL appears to achieve effective model calibration by balancing the distribution of confidence on the +output. Figure 5 plots the top 10 highest-probability classes for the best-performing class for each +training method for CIFAR100. Not only does the training methodology change the best performing +class, but the distribution of output confidence for each class is also very different. The results of +over-softening of label smoothing and over-confidence in the one-hot label are also observed here. +Figure 6 shows examples of soft labels obtained from the teacher network after the input sample +undergoes data augmentation in training on the ImageNet. We observed that the distribution of labels +supervising student networks differed significantly from that of the CutMix. +Penultimate layer output visualization. We plot the activation of the penultimate layer to visualize +the effect of LDL on the feature representation as in [18, 19]. ’beaver’ and ’otter’ are semantically +similar classes and ’dolphin’ are semantically different classes. Very interestingly, it is observed +that LDL plays an appropriate role in the classification of semantically similar classes. Looking +at the first row of Figure 7, the online LDL has a geometry of activation that is more effective for +semantically similar classification when long training is performed. Similarly, for the relationships of +’man’, ’woman’, and ’sunflower’, although less prominent than in the previous example, online LDL +produces more efficient geometries for classification than other methods. +8 + +100 +score for each class +Vanilla(78.11/0.04) +90 +Label Smoothing(78.68/0.04) +KD(77.91/0.04) +CutMix(80.63/0.04) +Off-LDL(78.86/0.04) +80 +On-LDL(81.22/0.04) +On-LDL+(81.51/0.04) +70 +score < 60 +60 <= score < 70 +70 <= score < 80 +60 +80 <= score < 90 +90 <= score +50 +0.40.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +confidence for each class100 + score for each class +Vanilla(78.55/0.038) +90 +Label Smoothing(78.78/0.038) +KD(78.67/0.038) +CutMix(81.85/0.038) +Off-LDL(78.81/0.038) +80 +On-LDL(83.39/0.038) +On-LDL+(83.62/0.038) +70 +score < 60 +60 <= score < 70 +60 +70 <= score < 80 +80 <= score < 90 +90 <= score +50 +0.40.45 +0.5 +0.55 +0.6 +0.650.7 +0.75 +0.8 +0.85 +0.9 +0.95 +confidence for each classResNet50 +ResNet18 +Figure 5: The top 10 high-probability classes according to the best-performing classes by train- +ing methodology. For each method, the F1 score for the best performing class was recorded next to +the class name and overall accuracy was recorded next to the method name. +CutMix: nipple (0.68), leopard (0.32) +Online LDL: mud turtle (0.05), library (0.19) +Online LDL: nipple (0.23), leopard (0.06) +CutMix: mud turtle (0.31), library (0.68) +Figure 6: Examples of supervision by CutMix and LDL. These examples from ImageNet are label +distribution and input sample pairs obtained from a teacher network after CutMix augmentation. +Vanilla +Label Smoothing +KD +Off-LDL +CutMix +On-LDL +On-LDL+ +Figure 7: Visualization of penultimate layer’s activation. The first row is the training samples of +the ’beaver’, ’otter’, and ’dolphin’ classes, and the second row is the test samples. The third row +is the training samples of the ’man’, ’woman’, and ’sunflower’ classes, and the last row is the test +samples. All plots are visualization of the activation of ResNet50. +5 +Conclusion +We observed and analyzed the effects of label smoothing, KD, and data augmentation on classification +accuracy and model confidence in terms of label distribution learning. Although the current approach +has limitations in that the method for generating labels is relatively simple and limited to image +classification, through experiments and visualization on four data sets, the LDL-based training +approach can simultaneously improve model accuracy and calibration. As a future study, we plan to +verify the utility of LDL in other tasks and analyze LDL based on theoretical backgrounds such as +class-considered approaches and risk minimization [41]. +9 + +tank(95.96) +CutMix(81.85 +0.0030 +0.0025 +Probability +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +2 +tor +E +Classsunflower(94.36) +0.0040 +Label Smoothing(78.68) +0.0035 +0.0030 +Probability +.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Classawn.mower(95.43 +Vanilla(78.11) +0.0035 +0.0030 +Probability +0.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +cle +truck +motor +C +trad +Classbicycle(95.92) +On-LDL+(81.51) +Vanilla +0.008 +Label Smoothing +KD +0.006 +Probability +CutMix +Off-LDL +0.004 +On-LDL +On-LDL+ +0.002 +0.000 +Classsunflower(96.48) +0.0040 +On-LDL(81.22) +0.0035 +0.0030 +Probability +0.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +6 +SW +Classsunflower(96.45) +0.0040 +Off-LDL(78.86) +0.0035 +0.0030 +. +.0025 +Probabilit +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Classmotorcycle(94.12) +0.0035 +KD(78.67) +0.0030 +0.0025 +Probability +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +ke +cra +awn_ +Classsunflower(93.94) +0.0030 +LabelSmoothing(78.78) +0.0025 +Probability +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Classlawn mower(93.88) +Vanilla(78.55） +0.0030 +0.0025 +Probability +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Classpickup truck(96.52) +On-LDL+(83.62) +0.007 +0.006 +Probability +0.005 +0.004 +0.003 +0.002 +0.001 +0.000 +Classsunflower(98.02) +0.0030 +On-LDL(83.39) +0.0025 +Probability +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Classskyscraper(94.06) +Off-LDL(78.81) +Vanilla +0.0030 +Label Smoothing +0.0025 +KD +CutMix +Probability +0.0020 +Off-LDL +0.0015 +On-LDL +On-LDL+ +0.0010 +0.0005 +0.0000 +Classsunflower(96.97) +0.0040 +CutMix(80.63) +0.0035 +0.0030 +Probability +0.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +fish +Classsunflower(94.42) +0.0040 +KD(77.91) +0.0035 +0.0030 +Probability +1 +.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +Class1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +turtle +lizard +rth +box +monaste +cowboy boot +alligator I1.0 +0.8 - +0.6 +0.4 +0.2 +0.0 +bir +bath +bathing +remote +potter'sbeaver(99.9) +dolphin(99.8) +otter(99.5) +2 +0 +-2 +-4 +-10 +-5 +0 +5 +102.0 +beaver(64.49) +dolphin(77.32) +1.5 +otter(54.63) +1.0 +0.5 +0.0 +0.5- +-1.0 +-1.5 +2.0 +-4 +22.0 +beaver(66.08) +dolphin(71.2) +1.5 +otter(53.77) +1.0 +0.5 +0.0 +0.5 +-1.0 +1.5 +2.0 +-4 +2 +0 +42.0 +beaver(71.63) +dolphin(76.1) +1.5 +otter(61.61) +1.0 +0.5 +0.0 +0.5 +-1.0 +-1.5 +2.0 +4 +2 +0 +42.0 +beaver(72.3) +dolphin(80.2) +1.5 +otter(66.67) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-42.0 +beaver(66.36) +dolphin(73.63) +1.5 - +otter(51.78) +1.0 - +0.5 +0.0 - +-0.5 +-1.0 +1.5 +2.0 +4 +-2man(99.6) +woman(99.5) +sunflower(100.0) +0 +-2 +-4 +-10 +-5 +0 +5 +10man(100.0) +woman(100.0) +sunflower(100.0) +2 +-2 +-10 +-5 +0 +10man(100.0) +woman(100.0) +sunflower(100.0) +0 +-2 +-4 +-10 +-5 +0 +5 +10man(100.0) +woman(100.0) +sunflower(100.0) +0 +-2 +-4 +-10 +-5 +0 +5 +10man(97.18) +woman(97.31) +sunflower(100.0) +-2 +-10 +-5 +0 +5 +10beaver(99.9) +dolphin(100.0) +otter(99.7) +-2 +-10 +-5 +10man(99.2) +woman(99.0) +sunflower(99.9) +-2 +-4 +10 +-5 +0 +5 +10man(100.0) +woman(100.0) +sunflower(100.0) +-2 +-10 +5 +0 +： +102.0 +man(56.57) +woman(59.72) +1.5 +sunflower(93.0) +1.0 +0.5 - +0.0 +0.5 - +-1.0 +-1.5 +2.0 +-4 +-22.0 +man(62.94) +1.5 +woman(72.91)） +sunflower(96.08) +1.0 +0.5 +0.0 +0.5 +-1.0 - +-1.5 +2.0 +-4 +02.0 +man(66.67) +woman(67.66) +1.5 +sunflower(98.02) +1.0 +0.5 +0.0 +0.5 +1.0 +-1.5 +2.0 +-4 +22.0 +man(60.7) +woman(64.04) +1.5 +sunflower(92.39) +1.0 - +0.5 +0.0 +0.5 - +-1.0 +1.5 +2.0 +-4 +2 +02.0 +man(56.08) +woman(63.64) +1.5 +sunflower(93.94) +1.0 +0.5 +0.0 +0.5- +1.0 - +1.5 - +2.0 +-4 +-22.0 +man(61.0) +woman(62.5) +1.5 +sunflower(92.61) +1.0 +0.5 +0.0 +-0.5- +1.0 +1.5 +2.0 +-4 +-2 +0 +42.0 +man(63.77) +woman(67.36) +1.5 +sunflower(95.92) +1.0 +0.5 +0.0 +0.5 - +1.0 +1.5 +2.0 +-4 +-2 +0beaver(99.9) +dolphin(100.0) +otter(99.7) +2 +0 +-2 +-4 +-10 +-5 +5 +10beaver(99.8) +dolphin(100.0) +otter(99.7) +2 +0 +-2 +-4 +-10 +-5 +0 +5 +10beaver(98.59) +dolphin(99.1) +otter(98.3) +-2 +-10 +-5 +0 +10beaver(99.4) +dolphin(99.9) +otter(99.3) +2 +0 +-2 +-4 +-10 +-5 +0 +5 +10beaver(99.9) +dolphin(100.0) +otter(99.7) +-2 +10 +-5 +0 +5 +102.0 +beaver(68.81) +dolphin(74.47) +1.5 +otter(67.36) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +02.0 +beaver(62.78) +dolphin(73.37) +1.5 +otter(52.68) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4References +[1] Krizhevsky, A., Sutskever, I. & Hinton, G.E. 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In: +Proc. of CVPR. +Supplementary Materials +Overview +This supplementary material contains additional experiments and analyses not included in the main +manuscript. 1) Experimental results with uncertainty based LDL, 2) Additional model calibration +error measurement results, 3) Loss curve and training tendency, 4) Additional visualization of +feature representation, 5) LDL evaluation results for each dataset according to the student/teacher +configurations, 6) Pseudocode of LDL, and 7) Reliability diagrams according to each method are +included. +6 +Uncertainty based LDL +In addition to data augmentation, we evaluated uncertainty-based LDL as an online LDL method +for generating labels during training. Uncertainty-based LDL adopts variational inference [42] on +teacher networks to generate random labels ˆz = 1 +N +�N +i=1 σ(θt,i(x)), where N is the total number of +teacher networks dropped from MCDO and σ is the softmax function. We set the drop parameters of +11 + +MCDO ρ to 0.2 and N to 20 through empirical experiment. We used the mean of the softmax output +for MC dropout-based variance inference as ˆz. +We have applied another method for uncertainty-based LDL. Overfitting-to-generalization-ratio +(OGR) is originally designed to prevent overfitting during mutlmodal training by defining the ten- +dency of the gradient of each modality during training as an adaptive loss [43]. We modified +the OGR as per class loss difference between validation and training loss and use it as an adap- +tive loss or as label distribution by normalized weighting one-hot label. The OGR is defined as +OGRk = +((Lval,N+n−Ltrain,N+n)−(Lval,N −LtrainN ))2 +Lval,N+n−Lval,N +, where given model parameters θ at epoch N, +L is average loss of each kth class over the fixed train set, and n is gap of epoch. We set n to 10. +Table 1 shows the results obtained with each approach. Uncertainty-based LDL methods were also +able to achieve mostly improved performance compared to vanilla models, but the improvement was +small compared to on/offline LDL methods. Nevertheless, in almost all configurations of MCDO, +a positive effect could also be achieved in model calibration. In the case of OGR, the gain was not +even greater than that of MCDO, but when OGR was applied to the LDL method, more improved +results were obtained in accuracy and model correction than when the class-wise loss was used. It +was confirmed that overconfidence occurs when OGR is used as a class-wise loss. We speculate +that the reason why MCDO and OGR are difficult to achieve as much performance improvement as +the online LDL technique is that they generate the label distribution through only the change of the +label without considering the data. Nevertheless, uncertainty-based LDL was still able to validate its +potential in accuracy and model calibration. +Table 6: Classification accuracy and ECE of the uncertainty based LDL and other approaches for the +CIFAR 100. MCDO [42] based LDL are marked with T and S according to the application of the +teacher and student networks. OGR [42] includes two approaches: each class OGR is reflected in +loss or LDL is reflected. +Teacher +ResNet200 +Student +ResNet18 +ResNet50 +ResNet200 +Vanilla +77.83±0.55/0.080 +78.57±0.35/0.107 +79.47±0.58/0.101 +Temperature Scaling [4] +77.83±0.55/0.039 +78.57±0.35/0.032 +79.47±0.58/0.029 +Label smoothing +78.95±0.06/0.085 +78.90±0.24/0.042 +79.59±0.42/0.037 +KD (α=0.1,T=3) [7] +77.73±0.16/0.077 +79.12±0.45/0.103 +80.10±0.23/0.100 +CRD+KD [12] +80.41±0.14/0.108 +80.34±0.07/0.118 +81.58±0.20/0.110 +WSL [16] +79.91±0.03/0.111 +80.25±0.12/0.114 +80.96±0.01/0.107 +MCDO [42] +78.02±0.03/0.010 +79.38±0.14/0.104 +79.80±0.23/0.110 +On-LDL (MCDO-T) +77.95±0.13/0.072 +78.26±0.98/0.092 +79.56±0.45/0.084 +On-LDL (MCDO-S, T) +78.06±0.34/0.091 +79.23±0.18/0.089 +80.64±0.12/0.077 +On-LDL (MCDO-T, CutMix) +80.82±0.04/0.059 +82.26±0.25/0.032 +83.61±0.11/0.047 +OGR [43] +77.79±0.15 / 0.370 +78.64±0.17 / 0.352 +79.18±0.18 / 0.352 +On-LDL (OGR) +78.47±0.27 / 0.077 +79.58±0.24 / 0.101 +79.36±0.32 / 0.107 +On-LDL +81.26±0.11/0.043 +83.09±0.05/0.030 +83.88±0.28/0.038 +On-LDL+ +81.76±0.25/0.034 +83.57±0.05/0.038 +84.44±0.13/0.039 +7 +Additional Model Calibration Error Measurement +In addition to the ECE in the experiments, different calibration error metrics include uncertainty +calibration error (UCE), negative log-likelihood (NLL), and Brier score [6, 7, 8]. Table 2 shows that +for most model calibration error measurement metrics, the LDL-based method achieves the lowest +error except for temperature scaling, which is simple normalization. In particular, the NLL and Brier +scores achieve the lowest error in all cases. In the case of MCDO-based LDL, improved results were +observed in ECE, but high errors were observed in the remaining metrics, and very high in NLL and +Brier. +8 +Loss Curve and Training Tendency +Figure 8 shows the loss curves according to ResNet50-based vanilla, label smoothing, and online +LDL methods in CIFAR100 and training trends according to accuracy. The loss curve plotted the +log loss of the top 5 and bottom 5 accuracy classes to investigate the training trends for each class. +The most notable feature in the trend of the loss curve is the largest difference between the top 5 +and bottom 5 losses for vanilla training. LDL and label smoothing suppress this tendency, and the +12 + +Table 7: Evaluation results according to model calibration error scores (ECE, UCE, NLL, and Brier) +with different teacher (T) and student (S) configuration in CIFAR 100. +T(R200)/S(R18) +Score Metric +acc +ece +uce +nll +Brier +Vanilla +77.83±0.55 +0.080 +0.106 +0.914 +0.125 +Temperature Scaling [4] +77.83±0.55 +0.039 +0.033 +0.892 +0.121 +Label smoothing +78.95±0.06 +0.085 +0.138 +0.964 +0.122 +KD (α=0.1,T=3) [7] +77.73±0.16 +0.077 +0.104 +0.905 +0.124 +CRD+KD [12] +80.41±0.14 +0.108 +0.135 +0.903 +0.128 +WSL [16] +79.91±0.03 +0.111 +0.137 +0.936 +0.131 +Off-LDL +78.67±0.14 +0.060 +0.086 +0.826 +0.118 +On-LDL (MCDO-S, T) +78.06±0.34 +0.091 +0.123 +1.918 +0.336 +On-LDL +81.26±0.11 +0.043 +0.078 +0.801 +0.112 +On-LDL+ +81.76±0.25 +0.034 +0.073 +0.693 +0.103 +T(R200)/S(R50) +Score Metric +acc +ece +uce +nll +Brier +Vanilla +78.57±0.35 +0.107 +0.137 +0.933 +0.131 +Temperature Scaling [4] +78.57±0.35 +0.032 +0.044 +0.815 +0.114 +Label smoothing +78.90±0.24 +0.042 +0.085 +0.939 +0.116 +KD (α=0.1,T=3) [7] +79.12±0.45 +0.103 +0.133 +0.914 +0.129 +CRD+KD [12] +80.34±0.07 +0.118 +0.143 +0.943 +0.132 +WSL [16] +80.25±0.12 +0.114 +0.139 +0.938 +0.129 +Off-LDL +79.19±0.37 +0.087 +0.116 +0.835 +0.122 +On-LDL (MCDO-S, T) +79.23±0.18 +0.089 +0.120 +2.871 +0.543 +On-LDL +83.09±0.05 +0.030 +0.064 +0.633 +0.095 +On-LDL+ +83.57±0.05 +0.038 +0.065 +0.621 +0.096 +T(R200)/S(R200) +Score Metric +acc +ece +uce +nll +Brier +Vanilla +79.47±0.58 +0.101 +0.128 +0.895 +0.126 +Temperature Scaling [4] +79.47±0.58 +0.029 +0.033 +0.782 +0.111 +Label smoothing +79.59±0.42 +0.037 +0.073 +0.907 +0.111 +KD (α=0.1,T=3) [7] +80.10±0.23 +0.100 +0.128 +0.873 +0.124 +CRD+KD [12] +81.58±0.20 +0.110 +0.134 +0.879 +0.124 +WSL [16] +80.96±0.01 +0.081 +0.109 +0.787 +0.125 +Off-LDL +80.60±0.30 +0.087 +0.116 +0.835 +0.122 +On-LDL (MCDO-S, T) +80.64±0.12 +0.077 +0.106 +3.848 +0.766 +On-LDL +83.88±0.28 +0.038 +0.067 +0.607 +0.094 +On-LDL+ +84.44±0.13 +0.039 +0.065 +0.591 +0.092 +tendency becomes more pronounced after the first drop in learning rate. LDL has the worst test +performance at the beginning of training but has the highest accuracy after the first drop in learning +rate. This tendency is also observed when the log loss for the top and bottom 20 classes is plotted. +We observed the effect of label distribution learning in preventing overconfidence by class, and then +we are considering an approach that can adaptively apply label distribution and data augmentation by +class. The trend of this loss curve is similarly observed in previous studies [9, 10], but we were able +to observe the cause more clearly in the log loss by class, and the online LDL makes a more firm +contribution to the generalization performance. +13 + +Figure 8: Log loss and accuracy plots (ResNet50) for top and bottom (5 and 20 classes) for +CIFAR100. The upper graph plotted the top and bottom 5 classes, and the bottom graph plotted 20 +classes. +14 + +80 +Vanilla +Label Smoothing +60 +On-LDL +40 +20 +0. +0 +50 +100 +150 +200 +250 +1.5 +Top20 Vanilla +1.0 +M +Bottom20 Vanilla +ss +0.5 +M +Top20 LS +9 +0.0 +Bottom20 LS +-0.5 +Top20 On-LDL +-1.0 +Bottom20 On-LDL +0 +50 +100 +150 +200 +250 +Epochs80 +Vanilla +Label Smoothing +60 +On-LDL +c +40 +A +20 +0 +0 +50 +100 +150 +200 +250 +Top5 Vanilla +1 +Bottom5 Vanilla +ss +Top5 LS +9 +Bottom5 LS +Top5 On-LDL +-1 +Bottom5 On-LDL +0 +50 +100 +150 +200 +250 +Epochs9 +Additional Visualization of Feature Representation +Figure 9 shows additional visualization of the activation of the penultimate layer of ResNet50 for +ImageNet dataset. Each plot consists of two semantically similar classes of ImageNet and one +semantically different class. From the first line in Figure 9: [’green snake’,’water snake’,’grown’], +[’golf ball’,’ping-pong ball’,’pot’], [’pizza’,’potpie’,’espresso’], [’leopard’,’snow leopard’,’goldfish’], +[’timber wolf’,’brush wolf’,’black bear’], [’persian cat’,’siamese cat’,’mink ’]. The first two classes +are semantically similar, and the last class is a semantically different class. Through penultimate layer +visualization, it is observed that the on/offline LDL technique generates an effective representation for +classifying semantically similar classes except the case of [’timber wolf’,’brush wolf’,’black bear’] +(5th row). +Vanilla +LS +KD +CutMix +Off-LDL +On-LDL +On-LDL+ +Figure 9: +Visualization of penultimate layer’s activation of ResNet50 for ImageNet. +From the top: +[’green snake’,’water snake’,’grown’], +[’golf ball’,’ping-pong ball’,’pot’], +[’pizza’,’potpie’,’espresso’], [’leopard’,’snow leopard’,’goldfish’], [’timber wolf’,’brush wolf’,’black +bear’], [’persian cat’,’siamese cat’,’mink ’]. The first two classes are semantically similar, and the last +class is a semantically different class. +15 + +2.0 +green snake(63.16) +water snake(65.38) +1.5 +gown(70.0) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +golf ball(89.8) +ping-pong ball(78.1) +1.5 +pot(73.04) +1.0 +0.5 - +0.0 +0.5 +-1.0 +1.5 +2.0 +-4 +-2 +02.0 +golf ball(91.84) +ping-pong ball(80.73) +1.5 - +pot(70.18) +1.0 - +0.5 +0.0 +0.5- +1.0 - +1.5 +2.0 +-4 +2 +02.0 +golf ball(91.84) +ping-pong ball(85.44) +1.5 +pot(69.03) +1.0 +0.5 +0.0 +0.5 +-1.0 +-1.5 +2.0 +-4 +0 +2 +42.0 +golf ball(92.78) +ping-pong ball(87.38) +1.5 - +pot(73.04) +1.0 - +0.5 - +0.0 - +0.5- +1.0 - +1.5 +2.0 +-4 +-2 +0 +22.0 +golf ball(89.58) +ping-pong ball(86.0) +1.5 - +pot(70.09) +1.0 - +0.5 +0.0 +0.5- +-1.0 - +1.5 +2.0 +-4 +2 +0 +22.0 +pizza(77.08) +potpie(66.67) +1.5 +espresso(68.97) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +22.0 +pizza(84.54) +potpie(69.31) +1.5 - +espresso(72.16) +1.0 - +0.5 +0.0 +0.5- +-1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +pizza(82.0) +potpie(67.37) +1.5 +espresso(70.21) +1.0 +0.5 - +0.0 - +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +22.0 +pizza(81.63) +potpie(73.68) +1.5 - +espresso(69.39) +1.0 - +0.5 +0.0 +0.5- +1.0 +1.5 +2.0 +-4 +2 +0 +22.0 +pizza(82.83) +potpie(72.73) +1.5 - +espresso(72.53) +1.0 - +0.5 +0.0 - +0.5- +-1.0 +1.5 - +2.0 +-4 +2 +0 +22.0 +green snake(66.67) +water snake(72.73) +1.5 - +gown(71.43) +1.0 - +0.5 - +0.0 - +0.5 +-1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +pizza(82.35) +potpie(72.73) +1.5 +espresso(70.33) +1.0 +0.5 - +0.0 +0.5 +-1.0 +1.5 +2.0 +-4 +-2 +0 +22.0 +pizza(83.17) +potpie(77.89) +1.5 - +espresso(73.12) +1.0 - +0.5 +0.0 +0.5- +1.0 - +1.5 +2.0 +-4 +2 +0 +22.0 +leopard(77.08) +snowleopard(89.8) +1.5 +goldfish(91.09) +1.0 +0.5 - +0.0 +0.5 +-1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +leopard(83.17) +snow leopard(92.78) +1.5 - +goldfish(90.91) +1.0 - +0.5 +0.0 - +0.5- +-1.0 +1.5 +2.0 +-4 +2 +0 +42.0 +leopard(85.71) +snow leopard(90.91) +1.5 +goldfish(93.07) +1.0 +0.5 +0.0 +0.5 +1.0 +-1.5 +2.0 +-4 +2 +0 +22.0 +leopard(82.69) +snowleopard(90.72) +1.5 - +goldfish(92.93) +1.0 - +0.5 +0.0 +0.5- +-1.0 +1.5 - +2.0 +-4 +-2 +0 +22.0 +leopard(83.17) +snow leopard(90.91) +1.5 - +goldfish(92.78) +1.0 - +0.5 +0.0 +0.5- +1.0 - +1.5 +2.0 +-4 +-2 +0 +22.0 +leopard(81.9) +snow leopard(90.91) +1.5 +goldfish(90.2) +1.0 +0.5 +0.0 +0.5 +-1.0 +-1.5 +2.0 +-4 +0 +22.0 +leopard(79.61) +snow leopard(90.72) +1.5 +goldfish(90.91) +1.0 +0.5 +0.0 +0.5 +-1.0 +-1.5 +2.0 +-4 +-2 +0 +42.0 +timber wolf(73.39) +brush wolf(66.67) +1.5 +black bear(83.81) +1.0 +0.5 - +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +02.0 +green snake(64.65) +water snake(70.8) +1.5 +gown(65.35) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +0 +22.0 +timberwolf(77.48) +brush wolf(70.1) +1.5 - +black bear(83.81) +1.0 - +0.5 +0.0 +0.5- +1.0 - +1.5 +2.0 +4 +-2 +0 +22.0 +timberwolf(75.25) +brush wolf(69.9) +1.5 +black bear(82.69) +1.0 +0.5 +0.0 - +0.5 +-1.0 +1.5 +2.0 +-4 +2 +0 +22.0 +timber wolf(71.7) +brush wolf(66.67) +1.5 - +black bear(85.44) +1.0 - +0.5 +0.0 +0.5- +-1.0 +-1.5 +2.0 +-4 +2 +0 +22.0 +timberwolf(75.93) +brush wolf(69.47) +1.5 - +black bear(86.0) +1.0 - +0.5 +0.0 - +0.5- +1.0 - +1.5 +2.0 +4 +-2 +0 +22.0 +timberwolf(76.92) +brushwolf(68.09) +1.5 +black bear(82.69) +1.0 +0.5 - +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +02.0 +timber wolf(78.1) +brush wolf(68.09) +1.5 - +black bear(83.33) +1.0 - +0.5 +0.0 +0.5- +-1.0 - +1.5 - +2.0 +-4 +-2 +0 +22.0 +persian cat(88.0) +siamese cat(84.91) +1.5 +mink(76.19) +1.0 +0.5 - +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +persian cat(90.0) +siamese cat(88.89) +1.5 - +mink(80.39) +1.0 - +0.5 +0.0 +0.5 - +1.0 - +1.5 - +2.0 +-4 +2 +0 +2 +42.0 +persian cat(92.16) +siamese cat(82.88) +1.5 +mink(77.06) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +02.0 +persian cat(88.68) +siamese cat(86.79) +1.5 - +mink(80.77) +1.0 - +0.5 +0.0 +0.5- +1.0 +1.5 +2.0 +-4 +2 +0 +22.0 +green snake(67.35) +water snake(67.83) +1.5 - +gown(67.37) +1.0 - +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +2 +0 +22.0 +persian cat(90.0) +siamese cat(84.91) +1.5 - +mink(80.77) +1.0 - +0.5 +0.0 +0.5- +-1.0 - +1.5 - +2.0 +-4 +2 +0 +22.0 +persian cat(89.32) +siamese cat(89.52) +1.5 +mink(84.0) +1.0 +0.5 - +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +22.0 +persian cat(91.09) +siamese cat(89.72) +1.5 - +mink(83.02) +1.0 - +0.5 +0.0 - +0.5 - +-1.0 - +1.5 +2.0 +-4 +2 +0 +42.0 +green snake(63.92) +watersnake(69.03) +1.5 - +gown(68.63) +1.0 - +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-4 +-2 +2 +42.0 +green snake(65.35) +water snake(67.24) +1.5 +gown(73.27) +1.0 +0.5 +0.0 +0.5 +1.0 +-1.5 +2.0 +-4 +0 +2 +42.0 +green snake(68.63) +water snake(75.68) +1.5 - +gown(64.0) +1.0 - +0.5 +0.0 +0.5- +1.0 +1.5 +2.0 +-4 +-2 +0 +22.0 +golf ball(87.76) +ping-pong ball(76.52) +1.5 +pot(70.59) +1.0 +0.5 - +0.0 - +0.5 +1.0 +1.5 +2.0 +-4 +-2 +0 +2 +42.0 +golf ball(92.78) +ping-pong ball(86.54) +1.5 - +pot(68.42) +1.0 - +0.5 +0.0 +0.5- +1.0 - +1.5 +2.0 +-4 +-2 +0 +210 +LDL Model Comparison +Figure 10 shows a comparison plot between the proposed LDL-based best-performing model and +other training methods. For each plot, the x-axis is classification accuracy and the y-axis is ECE. +Each graph shows the accuracy/ECE relationship for all student networks trained from a specific +teacher network. The CNN models with the LDL approach in all datasets are located in the lower +right of the graph, which means that the LDL achieves low ECE and high classification accuracy. +CIFAR100 +CIFAR10 +STL10 +ImageNet +Figure 10: Comparison of performance across specific datasets and teacher networks. The x- +axis is classification accuracy, and the y-axis is ECE. Each graph was plotted in different colors and +shapes according to the training methodology, and each model name is indicated in each figure. +16 + +T:R44 +0.12 +Vanilla +LS +resnet20 +KD +esnet32 +esnet110 +0.10 +resnet44 +Our best +resnet56 +esnet110 +0.08 +resneti +resnet110 +resnet +E +C +0.06 +0.04 +resneti +res +0.02 +resr +0.00. +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +AccuracyT:R110 +0.07 +Vanilla +LS +resne4et56 +KD +0.06 +resnet110 +Our best +resnet32 +resnet20 +0.05 +0.04 +110. +resh +eti10 +resesi +esne +C +resne +et20 +E +resnet20 +0.03 +0.02 +resnet4 +resi +0.01 +resnet2 +snets +0.00. +91 +92 +93 +94 +95 +96 +AccuracyT:R56 +Vanilla +0.14 +LS +pehe110 +KD +5f5 6 +SSKD +0.12 +resnet110 +CRD+KD +resnet56 +resnet20 +resnet44 +Our best +resnet5ssnet11( +resne +32 +esnet +0.10 +snet32 +resnet32 + 0.08 +re +net +0 +C +resnet20 +resnet20 +0.06 +resnet20 +esnet110 +0.04 +resnet32 +0.02 +re +resnet. +resnet56 +resnet44 +0.00 +68 +70 +72 +74 +76 +78 +AccuracyT:R110 +Vanilla +0.14 +resretdt 10 +LS +resnet56 +resnet20 +resnet44 +resnet110 +KD +snet56 +et56 +SSKD +0.12 +resnet20 +resnet56 +esnet32 +resnet110 +CRD+KD +esnet44 +res*4 +WSL +Our best +0.10 +resnetet3 + 0.08 +C +resnnet? +0.06 +resnet20 +resnet11 +0.04 +resnet32 +0.02 +res +resnet3 +resnet56 +resnet44 +0.00 +68 +70 +72 +74 +76 +78 +AccuracyT:R200 +0.200 +Vanilla +LS +0.175 +KD +CRD+KD +WSL +0.150 +resnext2$ +4x64d +Our best +0.125 +reset50 +t200 +res +t50 +lesnetz +snet200 +C +dla +res +0.100 +dla +t18 +resi +0.075 +resnet18 +reesreex22940604d +0.050 +4x64d +densenet121 +00 +rdlheltsmet200 +t121 +0.025 +0.000 +76 +78 +80 +82 +84 +86 +AccuracyT:R152 +Vanilla +resnet18 +0.10 +LS +KD +Our best +0.08 +resnet50 + 0.06 +resnet101 +C +resntt +re1 +0.04 +resnet15 +eshepst50 +esnet50 +101 +52 +0.02 +reettis +she +18 +0.00 +68 +70 +72 +74 +76 +78 +80 +Accuracy11 +Implementation of On/offline LDL +1 import +torch +2 import +torch.nn as nn +3 import +numpy as np +4 +5 criterion = nn. CrossEntropyLoss () +6 for images , labels in loader: +7 +# Generate +data +augments +Images +8 +images = Data_Aug(image , aug_method) +9 +10 +# Generate +soft +labels +11 +if method in [’Off -LDL’, ’On -LDL’]: +12 +with +torch.no_grad (): +13 +labels = TeacherModel(images) +14 +15 +logits = StudentModel(images) +16 +17 +# Cross +entropy +loss +18 +loss = criterion(logits , labels) +19 +20 def +Data_Aug(images , aug_method , alpha =1.0): +21 +lam = np.random.beta(alpha , alpha) +22 +I_x , I_y = images.size ()[2:] +23 +# shuffle +minibatch +24 +index = torch.randperm(images.size (0)) +25 +rand_image = images[index] +26 +if aug_method == ’mixup ’: # MixUp +Algorithm +27 +images = lam * images + (1 - lam) * rand_image +28 +elif +aug_method == ’ricap ’: #RICAP +Algorithm +29 +# draw a boundary +position (w,h) +30 +w = int(np.round(I_x * np.random.beta(alpha , alpha ))) +31 +h = int(np.round(I_y * np.random.beta(alpha , alpha ))) +32 +w_ = [w, I_x -w, w, I_x -w] +33 +h_ = [h, I_y -h, h, I_y -h] +34 +# select and crop four +images +35 +cropped_images = {} +36 +for k in range (4): +37 +index = torch.randperm(images.size (0)) +38 +x_k = np.random.randint (0, I_x - w_[k] + 1) +39 +y_k = np.random.randint (0, I_y - h_[k] + 1) +40 +cropped_images [k] = images[index ][:, :, x_k:x_k + w_[k], y_k:y_k + h_[k]] +41 +images = torch.cat( +42 +(torch.cat(( cropped_images [0], cropped_images [1]) , 2) +43 +torch.cat(( cropped_images [2], cropped_images [3]) , 2)), 3) +44 +elif +aug_method == ’cutmix ’: # CutMix +Algorithm +45 +cut_w = np.int(I_x * np.sqrt (1. - lam)) # cut rate +46 +cut_h = np.int(I_y * np.sqrt (1. - lam)) +47 +48 +cx = np.random.randint(I_x) +49 +cy = np.random.randint(I_y) +50 +51 +bbx1 = np.clip(cx - cut_w // 2, 0, I_x) +52 +bby1 = np.clip(cy - cut_h // 2, 0, I_y) +53 +bbx2 = np.clip(cx + cut_w // 2, 0, I_x) +54 +bby2 = np.clip(cy + cut_h // 2, 0, I_y) +55 +images [:, :, bbx1:bbx2 , bby1:bby2] = rand_image [:, :, bbx1:bbx2 , bby1:bby2] +56 +57 +return +images +17 + +12 +Reliability Diagram +Reliability diagram can intuitively visualize the correlation between model confidence and inference +accuracy based on a histogram [6]. Figure 11, 12, 13, and 14 show the reliability diagram for each +dataset and approach. The x-axis of the reliability diagram is a histogram of the confidence at a +specific interval, and the y-axis is the expected value of the accuracy for the inferred confidence. The +better the model correction effect is, the more the plot values match the linear lines for the x and y +axes [6]. +Vanilla +Label Smoothing +KD +Off-LDL (best) +On-LDL (best) +Figure 11: Reliability diagram for CIFAR10. Accuracy and ECE for the dataset are indicated for +each method. +18 + +T:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=92.45 +ECE=0.035 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=9B.90 +ECE=0.007 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.56 +ECE=0.038 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.36 +ECE=0.060 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.85 +ECE=0.037 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.83 +ECE=0.032 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.80 +ECE=0.013 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.77 +ECE=0.038 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.53 +ECE=0.061 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.90 +ECE=0.038 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=94.32 +ECE=0.020 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=92.16 +ECE=0.051 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=94.53 +ECE=0.003 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.97 +ECE=0.040 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.95 +ECE=0.060 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=94.06 +ECE=0.039 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=94.44 +ECE=0.020 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=94.92 +ECE=0.005 +0.9 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=92.53 +ECE=0.032 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.33 +ECE=0.019 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.18 +ECE=0.006 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.24 +ECE=0.038 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=9B.16 +ECE=0.053 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=9B.96 +ECE=0.020 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=9B.34 +ECE=0.038 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceVanilla +Label Smoothing +KD +Off-LDL (best) +On-LDL (best) +Figure 12: Reliability diagram for SLT10. Accuracy and ECE for the dataset are indicated for each +method. +19 + +T:R44. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=83.35 +ECE=0.067 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=86.96 +ECE=0.004 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.20 +ECE=0.073 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.78 +ECE=0.099 +0.%.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.22 +ECE=0.071 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=85.09 +ECE=0.057 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R44 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=86.71 +ECE=0.024 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.84 +ECE=0.075 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R56 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=84.35 +ECE=0.094 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.10 +ECE=0.072 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R56 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=84.71 +ECE=0.054 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.30 +ECE=0.109 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=87.22 +ECE=0.008 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=81.92 +ECE=0.087 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R110 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=81.80 +ECE=0.102 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.29 +ECE=0.077 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=85.45 +ECE=0.052 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=87.48 +ECE=0.006 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.16 +ECE=0.074 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.56 +ECE=0.052 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=86.72 +ECE=0.006 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.51 +ECE=0.067 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=82.75 +ECE=0.104 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=83.62 +ECE=0.079 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R44, S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=85.08 +ECE=0.050 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceVanilla +Label Smoothing +KD +Off-LDL (best) +On-LDL (best) +CRD+KD +WSL +Figure 13: Reliability diagram for CIFAR100. Accuracy and ECE for the dataset are indicated for +each method. +20 + +T:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=68.94 +ECE=0.074 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=70.75 +ECE=0.097 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=7B.60 +ECE=0.129 +08:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=73.86 +ECE=0.129 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=7B.43 +ECE=0.017 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=72.87 +ECE=0.053 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=72.19 +ECE=0.123 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=72.56 +ECE=0.015 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=72.63 +ECE=0.121 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=75.70 +ECE=0.125 +0.8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=75.85 +ECE=0.022 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=68.88 +ECE=0.047 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=74.99 +ECE=0.134 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R56 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=75.19 +ECE=0.069 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=7B.70 +ECE=0.136 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=73.53 +ECE=0.042 +08:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=73.43 +ECE=0.136 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=77.32 +ECE=0.119 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=75.90 +ECE=0.130 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=76.49 +ECE=0.014 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R110 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=77.55 +ECE=0.019 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=77.28 +ECE=0.084 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=68.80 +ECE=0.072 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R18 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=78.89 +ECE=0.085 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R18 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=77.57 +ECE=0.076 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R18 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=80.27 +ECE=0.109 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R18 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=79.94 +ECE=0.110 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=78.86 +ECE=0.060 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=82.00 +ECE=0.025 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=78.12 +ECE=0.105 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=78.66 +ECE=0.032 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=78.67 +ECE=0.102 +08:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=80.44 +ECE=0.117 +0%:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=71.33 +ECE=0.131 +0.8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=80.11 +ECE=0.116 +08:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=79.72 +ECE=0.084 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=8B.63 +ECE=0.037 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R200 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=79.06 +ECE=0.103 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R200 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=79.17 +ECE=0.032 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R200 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=79.83 +ECE=0.102 +0.8:0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R200 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=81.37 +ECE=0.111 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R200 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=80.95 +ECE=0.108 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200. S:R200 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=81.01 +ECE=0.081 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R200, S:R200 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=84.56 +ECE=0.037 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=71.33 +ECE=0.131 +0.8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=70.92 +ECE=0.017 +0.0 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110, S:R20 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=69.73 +ECE=0.061 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=71.01 +ECE=0.096 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R110. S:R32 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=71.13 +ECE=0.024 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceVanilla +Label Smoothing +KD +Off-LDL (best) +On-LDL (best) +Figure 14: Reliability diagram for ImageNet. Accuracy and ECE for the dataset are indicated for +each method. +21 + +T:R152. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=70.39 +ECE=0.014 +0.9 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=76.21 +ECE=0.034 +0.8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R101 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=77.43 +ECE=0.045 +00 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R101 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=78.36 +ECE=0.058 +0..0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R101 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=77.59 +ECE=0.046 +0.%.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R101 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=78.72 +ECE=0.023 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R101 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=79.09 +ECE=0.029 +080 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=70.41 +ECE=0.102 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=71.64 +ECE=0.013 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=70.52 +ECE=0.013 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R18 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=69.17 +ECE=0.086 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=75.94 +ECE=0.033 +0.9 +8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=77.53 +ECE=0.034 +0.0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152, S:R50 +1.0 +0.8 +0.6 +0.4 +0.2 +ACC=77.28 +ECE=0.021 +0.2 +0.4 +0.6 +0.8 +1.0 +ConfidenceT:R152. S:R50 +1.0 +Expected Accuracy +0.8 +0.6 +0.4 +0.2 +ACC=76.53 +ECE=0.069 +0.8.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence \ No newline at end of file diff --git a/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/load_file.txt b/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0600bb4f02c753fc1b9f1c4956fa0cbeb6a090e --- /dev/null +++ b/_dFQT4oBgHgl3EQf8DZn/content/tmp_files/load_file.txt @@ -0,0 +1,4037 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf,len=4036 +page_content='Rethinking Soft Label in Label Distribution Learning Perspective Seungbum Hong, Jihun Yoon, Bogyu Park, and Min-Kook Choi∗ AI Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Group Hutom Seoul, Republic of Korea mkchoi@hutom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='io Abstract The primary goal of training in early convolutional neural networks (CNN) is the higher generalization performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' However, as the expected calibration error (ECE), which quantifies the explanatory power of model inference, was recently introduced, research on training models that can be explained is in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We hypothesized that a gap in supervision criteria during training and in- ference leads to overconfidence, and investigated that performing label distribution learning (LDL) would enhance the model calibration in CNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To verify this assumption, we used a simple LDL setting with recent data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Based on a series of experiments, the following results are obtained: 1) State-of-the-art KD methods significantly impede model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 2) Training using LDL with recent data augmentation can have excellent effects on model cali- bration and even in generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 3) Online LDL brings additional improvements in model calibration and accuracy with long training, especially in large-size models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Using the proposed approach, we simultaneously achieved a lower ECE and higher generalization performance for the image classification datasets CIFAR10, 100, STL10, and ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We performed several visualiza- tions and analyses and witnessed several interesting behaviors in CNN training with the LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 1 Introduction The supervision of a convolutional neural network (CNN) using a hard label has been very successful in most image classification problems [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' However, in the training of a CNN using a hard label, as the number of weights of the network increases, [4] analyzed the overconfidence of the network prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To handle this phenomenon, [4] proposed the expectation of calibration error (ECE) to estimate the confidence of the model, and several approaches for calibrating the overconfidence of deep learning models were suggested, but they were not correlated with generalization performance and model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Recently, several studies have introduced that data augmentation is effective for model generalization as well as calibration [5, 6], but the results are not significant in terms of generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Label distribution learning (LDL) is designed for effective training through label distribution when the types of labels for supervision are difficult to define discretely and approaches the label generation (or enhancement) process as an optimization problem [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Typically, LDL has been applied to applications that include inherent label ambiguity, such as facial age estimation, head pose estimation, facial emotion estimation, multi-label learning, partial multi-label learning, and video summarization [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' From a LDL perspective, label smoothing [14, 17, 19] is considered a subset of LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='We ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='13444v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='CV] 31 Jan 2023 cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog person cat dog cat dog person cat dog person cat dog person cat dog person person Figure 1: The difference between the supervision using the hard label (blue box) and using the soft label (red box) for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' An illustration of the LDL for the cat class at the left is shown, and the reliability diagrams at the right show comparisons of the traditional hard label-based and LDL-based classification accuracy and ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Our LDL-based training successfully achieved better classification accuracy and lower ECE simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' are inspired by the basic concepts of LDL and assume that the LDL potentially overcomes the discrepancy between the one-hot label-based training and the maximum confidence- based testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To introduce this idea in a simple way, we exploited soft labels from teacher networks as a baseline distributed label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To learn the label distribution online differ from the former optimization approach, recent data augmentation techniques that merge labels and data during training were simultaneously applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' By applying the on/offline label distribution learning scenarios, we simultaneously obtained an improvement in the model generalization and calibration without additional regularization or archi- tecture modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The left section of Figure 1 shows examples of the difference between hard label and LDL–based supervision for cat recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The graphs at the right of Figure 1 show the reliability diagram [4] when different settings of modern KD approaches in the same CNN model are applied for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To verify the strength of LDL for model generalization and calibration, we performed a series of image classification tasks on datasets such as CIFAR10, 100 [23], STL10 [24], and ImageNet [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Based on a series of experiments with image classification, we confirmed that most recent KDs cause severe overconfidence, which impedes model calibration, and even a simple LDL approach can achieve better classification accuracy and suppression of model overconfidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 2 Related Works Model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ECE is an error that measures whether the prediction of the neural network can accurately estimate the true likelihood of the input data of the trained classes [4, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In [4], temperature scaling was proposed in a way that can effectively be corrected and the reliability diagram is used to visualize model confidence for CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In [30], various structural dropout methods and experiments on the drop rates according to each method were applied to the CNN model to analyze the correlation between model accuracy and ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' VWCI [31] reduced the ECE and improved the recognition performance by defining a confidence integration loss as a probabilistic regularization term defined from a Bayesian model using multiple inferences based on probabilistic depth and dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' It has also been reported that the AvUC loss based on uncertainty estimation in the model also aids in model calibration [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In addition to this, model training with mixup augmentation has been demonstrated to be effective in model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' However, it did not achieve much in improving the generalization performance of the model in preparation for the correction effect [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Label smoothing and label distribution learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Label smoothing was proposed to soften the hard label in the training process according to the given coefficient to prevent overconfidence and improve generalization performance [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In [18], the authors analyzed the effect of label smoothing on deep neural network training by visualizing the penultimate fully connected layer of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' According to the analysis results, there is evidence that the trained teacher network applied with label smoothing in the KD scenario can invalidate the effect of student model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In recent studies [19, 28], the effect of label smoothing on teacher networks in the KD scenario was analyzed in more detail to extend the research results [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In [19], a quantification method that label smoothing erases meaningful information in the teacher network logit was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In [28], the relationship between KD and label smoothing from the bias-variation perspective was analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 2 ReliabilityDiagram(T:R110,S:R56) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 Expected Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='4 Vanilla(72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='19) TS(72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 LS(72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='56) KD(72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='63) CRD+KD(75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='7) WSL(74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='99) Ourbest(75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 ConfidenceReliabilityDiagram(T:R200,S:R18) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 Expected Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='4 Vanilla(77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28) TS(77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 LS(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='89) KD(77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='57) CRD+KD(80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27) WSL(79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='94) Ourbest(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 ConfidenceOFrom the LDL perspective, Label smoothing can be regarded as a possible solution for LDL through constant softening of the hard label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We included label smoothing in our comparisons as one of the baselines to suppress overconfidence [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Since the introduction of the KD [7], a vast number of approaches for knowledge distillation have been proposed [8-16, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' FitNet [8] proposed a KD method that makes the feature maps of teacher networks similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In recent years, various approaches have been proposed from the perspective of representation learning, such as RKD [10], which achieved transfer learning through geometric relations to the output of the model, CRD using metric learning [12], mutual learning based [13], self-supervised learning–based KD [15], and weighted soft label-based KD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Among the variations of KD, the born-again network [9], which achieves transfer learning through repetitive training of student models without the use of a teacher network, and similar to [9], variations of KD that are free from teachers [14, 29] were also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' [18] and [19] explain the relationship between KD and label smoothing of teacher networks through empirical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We argue that KD should be described in terms of LDL rather than label smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We have observed that modern KDs spoil model calibration to improve generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 3 On/Offline Label Distribution Learning (LDL) In this section, we briefly introduce the notations and approaches for KD and label smoothing (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1), which are basic prerequisites for our LDL-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Subsequently, on and offline approaches for LDL are described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1 Preliminaries Knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' When the weight w of the last fully connected layer for the ith feature input x and the output with the softmax function for the kth class is given as pk i = e(xi)T wk �L l=1 e(xi)T wl , the softening output for the neural network is given by [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ¯pk i (x) = e((xi)T wk)/τ �L l=1 e((xi)T wl)/τ , (1) where τ is a temperature scaling parameter that determines size of softening, and the total loss function L = (1 − λ)LCE(y, pθs) + λLCE(¯pθt, ¯pθs) for the teacher model θt and the student model θs, where y is one-hot label and LCE(p, y) = − �K k=1 yk log(pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Label smoothing The label smoothing for the hard label yi for the same feature input is given as follows: ˜yk i = (1 − α)yk i + α K − 1, (2) where α is the smoothing coefficient, and the probabilities for each class except the hard target corresponding to the kth class are evenly distributed as α/(K − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Expected calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The ECE for estimating the confidence of the neural networks proposed in [4] is estimated for Ntest samples when the softmax output pi inferred for all test data and the index with the maximum probability in the output is ˆci = argmax i (pi = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ECE = M � m=1 |Hm| Ntest � 1 |Hm| � i∈Hm 1(ˆci = ci) − pi � , (3) where Hm is the index set and generates M interval bins of ((m − 1)/M, m/M] for Ntest samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Typically ECE is measured by the histogram for a bin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1 size by setting M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Criterion of LDL with cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The main objective of cross entropy loss with LDL perspective is given by: 3 ( ) To Be Learned hard label output ① Vanilla To Be Learned soft label output Smarter Model KD To Be Learned output soft label Smarter Model ③ Offline LDL To Be Learned output soft label Smarter Model 1 Smarter Model k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ④ Offline-En- LDL To Be Learned output soft label Smarter Model 2 Smarter Model 1 Smarter Model k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' output soft label output soft label ⑤ Offline-MT- LDL output Smarter Model To Be Learned ⑦ output ⑥ To Be Learned Label smoothing output LS soft label ② ⑧ Online LDL 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 𝑥" "𝑥 ⑨ To Be Learned output soft label Online-En-LDL 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 𝑥" "𝑥 Smarter Model 1 Smarter Model k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To Be Learned output soft label Online-MT-LDL 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 𝑥" "𝑥 Smarter Model 1 Smarter Model k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' output soft label hard label Figure 2: Schematic of the training configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We used abbreviations to simplify the notation: #1 learning from scratch (Vanilla), #2 label smoothing (LS), #3 knowledge distillation (KD), #4 soft label (Off-LDL), #5 teacher ensemble for soft label (Off-En-LDL), #6 linear combination with the multiple soft label (Off-MT-LLD), #7 soft label with data augmentation (On-LDL), #8 soft label using data augmentation with teacher ensemble (On-En-LDL), #9 linear combination of multiple soft label using data augmentation (On-MT-LDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The red box represents the existing training method, the green box represents the offline approaches, and the blue box represents the online approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' LCE(p, z) = − K � k=1 zk log(pk), (4) where z is a label vector that satisfies � j zj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Label smoothing or soft label by output of teacher networks can be regarded as a specific solution for z (z = ˜y(α) or z = ¯pk θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We reformulated the problem argmaxθ(E[hDtrain(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θ)] − E[hDtest(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θ)]) to find the CNN with the maximum generalization performance in a specific image classification dataset D ∋ {Dtrain, Dtest} as follows: argmax θ,Z E[hDtrain(x, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θ)] − E[hDtest(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θ)], (5) where ZDtrain ∋ {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=', zNtrain} is a set of new labels for all training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Equation (5) can be solved as an optimization problem of finding a pair of the optimal labels for each input data (xi, zi) such as the previously proposed LDL approaches [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Since traditional approaches cannot update z and θ simultaneously during deep neural network training, we applied simple but effective on/offline approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For the simplicity, we used basic KD setting, which teacher network generate new label set Z for target (student) neural network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The Offline setting is a way to generate Z as a soft label as the output of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Z is not updated during training, but some variations are possible depending on the way the ensemble of teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The online setting is a way of continuously transforming Z while updating θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Since it is difficult to presume an optimal Z, we generated diverse labels with modern data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' As with offline settings, there are several variations depending on the way the ensemble of teacher networks and label generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 2 shows different types of training configurations for baseline and LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 Offline LDL We simplified the problem by using the KD setting by the teacher output as the new label to extract feasible solutions for each sample pair (xi, zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The offline LDL is illustrated in the green box in Figure 2, such that the set of sample pairs (XDtrain, ZDtrain) under XDtrain ∋ {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=', xNtrain} is fixed during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The cross-entropy loss for training with the label generated by the teacher θt and the student model to be trained is θs is as follows: LCE(p, ¯z) = − K � k=1 ¯zk i log(pk i,θs), (6) where ¯z is defined by way of generating new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We used simple variations of ¯z = f(·) for offline LDL (see Figure 2): soft label f(xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θt) (Off-LDL, #4), soft label with teacher ensemble ¯z = 1 N �N n=1 f(xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θt,n), where N is the number of teacher models (Off-En-LDL, #5), and linear combination of soft labels from multiple teachers − �N n=1 �K k=1 ¯zk i,θt,n log(pk i,θs) (Off-MT-LDL, #6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='3 Online LDL To reflect the objective of Equation (5) during training, it is necessary to update Z and θ simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We applied recent data augmentation techniques that can simply adopt online LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Among the proposed data augmentation techniques, some techniques manipulate input data and label together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' A generalized form of augmentation technique considering data and labels together is ˆx = M1 ⊗ x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' + MP ⊗ xP ˆy = λ1y1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' + λP yP , (7) where ˆx is an augmented sample of mixed label ˆy with �P λi = 1 up to P samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' M is a blending mask equal to the data width and height, and satisfies �P Mi(u, v) = 1, where u and v indicate the pixel location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Each augmentation algorithm is designed to stochastically determine the location and size of each sample for blending and mainly follows a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' M is provided differently for each augmentation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Typically, When P = 2, x1 is defined as the target image, and x2 is defined as the 0 image for CutOut [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We exploited mixup [33], CutMix [34], and RICAP [35] for data augmentation, and online LDL with data augmentation was as follows: LCE(ˆp, ˆz) = − K � k=1 ˆzk i log(ˆpk i,θs), (8) where ˆpk i = e(ˆxi)T wk �L l=1 e(ˆxi)T wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Data augmentation applies equally to teacher models for label en- hancement ˆzk i , but the mixed label ˆy is not used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Similar to the offline approaches corresponding to #5 and #6 in Figure 2, online LDL can be easily extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The enhanced label through the augmentation-based teacher ensemble is obtained as ˆz = 1 N �N n=1 f(ˆxk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' θt,n) (On-En- LDL, #8), and the linear combination of the augmentation-based soft labels from multiple teachers is given as − �N n=1 �K k=1 ˆzk i,θt,n log(ˆpk i,θs) (On-MT-LDL, #9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 4 Experimental Results Experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We performed a series of experiments on the CIFAR10, 100 [23], and STL10 [24] datasets to verify the performance of the on/offline LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' First, the major experiments were performed with the teacher-student network configurations of the well-used ResNet architectures [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We divided ResNet into small and large networks according to model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Small size models include ResNet20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27M), 56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85M), and 110 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='7M) and large size models include ResNet18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='18M), 50 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='51M), and 200 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='62M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For CIFAR10, 100, and STL10, a total of 240 epochs was trained to start with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1 learning rate scaling was applied at 150, 180, and 210 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The weight decay was set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 × 10−4, the batch size was set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We also tested a long training scenario with a basic training configuration, with the assumption that LDL can fundamentally achieve better performance when the number of training pairs of sample and label is large especially in online LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For long training, a total of 350 epochs was trained to start with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1 learning rate scaling was applied at 150, 200, 250, and 300 epochs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In all ensemble (En) and multiple teacher (MT) settings, ResNet20, 32, 44, 56, and 110 used together, were trained by the same learning scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We measured the image classification accuracy and ECE [4] to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Visualization of reliability diagrams is provided to intuitively check the strength of model calibration of the network in the same way as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' LDL with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The augmentation algorithms applied for On-LDL are mixup [33], CutMix [34], and RICAP [35], and the default hyper-parameter for data augmentation refers to the original implementation of each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 1 shows the recognition results of the on/offline LDL according to each data augmentation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' All three algorithms showed rather poor performance in small networks and were able to achieve significant performance improvement in large networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For long training, we only report the LDL of the augmentation technique with the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' All training was performed on 3 different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We omit variations in 2The long training is marked with ’+’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 5 Table 1: Classification accuracy (%) and ECE of vanilla and each LDL setup for CIFAR100 depending on each data augmentation methods [33, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Teacher ResNet110 ResNet110 ResNet200 ResNet200 Student (# param) ResNet20 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27M) ResNet56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85M) ResNet18 (11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='008 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='010 calibration scores, which are no significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We selected the CutMix [34] or RICAP [35] as a data augmentation for remain LDL experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Similar to the results of [5], with the mixup augmentation, the improvement in accuracy was not significant, but a model calibration effect could be seen in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Rather, CutMix and RICAP achieved steady performance improvement and better model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Offline LDL achieved the best accuracy and model calibration performance in ResNet20, and online LDL improved significantly as the size of the network increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' With the ResNet50, an accuracy improvement of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='9% and better model calibration was achieved than in the case of data augmentation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' CIFAR10 and STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 2 shows the evaluation results of the CIFAR10 and STL10 datasets for LDL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In the two relatively small datasets, we did not evaluate the large network, as only the small network could provide sufficient performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In both CIFAR10 and SLT10, online LDL utilizing multiple teacher networks showed improvements in accuracy and model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In CIFAR10, the accuracy increase of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6% and the ECE reduction effect of up to about 80% were obtained in ResNet56 compared to the vanilla model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In STL10, the accuracy increase of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='39% and ECE improvement effect of about 85% were obtained in ResNet20 compared to the vanilla model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In CIFAR10 and STL10, RICAP achieved better performance than CutMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 3 shows the evaluation results of LDL in CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In the upper part of Table 3, the improvement in generalization performance is relatively insignificant for small networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We hypothesized that a large number of weights is required to sufficiently train the LDL-based label variants and evaluated the small and large networks simultaneously on the CIFAR100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' When the model has a large number of weights, it is well trained on the label variation of the new label distribution, and it is possible to achieve sufficient performance improvement and model calibration without the multiple teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We obtained two main observations from the experiments with three benchmarks: 1) The classification accuracy is mainly determined by the number of weights with LDL approaches, and the influence of the number of parameters in the teacher model is not noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 2) The effectiveness of model calibration steadily improves regardless of the number of parameters in each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We also compared the proposed LDL approaches with the SOTA KD methods [12, 15, 16] on the CIFAR100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 3 shows the comparison results with small and large student networks in terms of classification accuracy and ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In most cases, LDL-based methods achieved a higher generalization performance and lower ECE simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 3 shows this phenomenon more dramatically: as the number of parameters in the student network is small, the SOTA KD methods show a very high ECE score, have no explanatory power for model confidence, 6 Table 3: Classification accuracy and ECE of the LDL and the SOTA KDs for the CIFAR 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Among the LDL techniques, the performance of the models that achieved the highest accuracy or the lowest ECE were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Teacher ResNet110 Student ResNet20 ResNet56 ResNet110 Vanilla 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='131 Label smoothing 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='20/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='053 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='17/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='020 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='16/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='051 KD (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1,T=3) [7] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='23/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='071 72.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 Figure 3: Performance change by methods according to small and large student models for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The left graph shows the accuracy difference with the vanilla training by the number of weights of the student model, and the right graph shows the ECE difference with the vanilla training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The LDL technique shows continuous improvement in model correction as the accuracy increases, and the effect increases as the model size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' On the other hand, SOTA KD methods have a steady improvement in accuracy, but rather impede model reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' and result in overconfidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' LDL techniques were able to achieve better model calibration compared to SOTA KDs as the number of parameters in the student network increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Classification accuracy showed up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='86% improvement in ResNet200 compared to the best performing CRD+KD, and an improvement of at least 60% up to 88% compared to the SOTA KD methods in model calibration3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We evaluated the ImageNet dataset [25] to verify LDL methods on the large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For the evaluation of the ImageNet, we set up the multiple teacher and student network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The learning scheduler applied the configuration of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=', and the on/offline LDL methods were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' As shown in Table 4, similar to the other datasets, the similar tendency of improved accuracy and model calibration was achieved simultaneously compared to vanilla and label smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Other model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' To validate the effect of LDL in various architectures, we performed an evaluation according to ResNet200 teacher and ResNeXt [36], DenseNet [37], and DLA [38] student networks in the CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 5 lists the accuracy and model calibration improvements by LDL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Compared to the vanilla model in other types of architectures, there was an accuracy improvement of about 3-5% and an ECE reduction of 20-66% was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 3Types of teacher networks, long training results of SOTA KD methods, loss curves, and additional experi- mental results and analysis are in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 7 LDL (ours) CRD+KD [12] WSL [16] 4 3 2 1 0 ResNet20 ResNet56 ResNet110LDL (ours) - CRD+KD ［12] WSL[16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 ResNet20 ResNet56 ResNet110LDL (ours) ■ CRD+KD [12] WSL [16] 5 4 3 2 1 0 ResNet18 ResNet50 ResNet200LDL (ours) CRD+KD [12] WSL [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 ResNet18 ResNet50 ResNet200Table 4: Classification accuracy (top1 / top5) and ECE for ImageNet dataset depending on each label enhancement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Teacher ResNet152 ResNet152 ResNet152 ResNet152 Student ResNet18 ResNet50 ResNet101 ResNet152 Vanilla 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='39/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='014 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='96/92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='81, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='032 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='43/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='72, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='045 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='050 Label smoothing 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='40/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='102 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='56/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='070 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='36/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='058 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='82/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='036 Off-LDL 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='63/90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='46, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='013 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='61, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='021 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='72/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='024 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='16/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='021 On-LDL 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='03/88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='91, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='078 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='84/92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='017 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='97/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='016 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='51/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='72, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='022 On-LDL+ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='43/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='062 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='30/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='023 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='82/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='022 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='25/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='021 Table 5: Evaluation of different types of architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Method Student (# param) ResNext29_4x64d (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1M) [36] DenseNet121 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='95M) [37] DLA (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='29M) [38] Vanilla 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='050 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='26/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='081 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='50/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='099 Label smoothing 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='37/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='151 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='044 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='33/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 KD (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1,T=3) [7] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='34/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='052 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='080 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='21/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='101 Off-LDL 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='20/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='062 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='081 On-LDL 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='059 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='21/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='028 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='033 ResNet18 ResNet50 Figure 4: Plotting the relationship between the F1 score and the average output confidence for the GT class of the model according to the training methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The average output confidence for each of the 100 classes of CIFAR100 and different shapes are marked for each F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Values in parentheses are the accuracy and ECE of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Why does LDL work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Based on the evaluation results of four datasets, LDL is observed to have an excellent effect on generalization performance and model calibration for CNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 4 is the result of the test set plotting the average confidence of ResNet18 and 50 for the ground truth class on the x-axis and the F1 score for each class on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In the case of one-hot label-based training or offline LDL using only soft labels of a teacher trained on the one-hot label, most have an average confidence score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='9 or higher regardless of the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Applying data augmentation or label smoothing alleviates this over-confidence, but in the case of label smoothing, the confidence distribution has an excessively large variance as a result of the forced softening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The case of online LDL appears to achieve effective model calibration by balancing the distribution of confidence on the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 5 plots the top 10 highest-probability classes for the best-performing class for each training method for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Not only does the training methodology change the best performing class, but the distribution of output confidence for each class is also very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The results of over-softening of label smoothing and over-confidence in the one-hot label are also observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 6 shows examples of soft labels obtained from the teacher network after the input sample undergoes data augmentation in training on the ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We observed that the distribution of labels supervising student networks differed significantly from that of the CutMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Penultimate layer output visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We plot the activation of the penultimate layer to visualize the effect of LDL on the feature representation as in [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' ’beaver’ and ’otter’ are semantically similar classes and ’dolphin’ are semantically different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Very interestingly, it is observed that LDL plays an appropriate role in the classification of semantically similar classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Looking at the first row of Figure 7, the online LDL has a geometry of activation that is more effective for semantically similar classification when long training is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Similarly, for the relationships of ’man’, ’woman’, and ’sunflower’, although less prominent than in the previous example, online LDL produces more efficient geometries for classification than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 8 100 score for each class Vanilla(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='11/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) 90 Label Smoothing(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='68/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) KD(77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='91/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) CutMix(80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='63/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) Off-LDL(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='86/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) 80 On-LDL(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) On-LDL+(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='51/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04) 70 score < 60 60 <= score < 70 70 <= score < 80 60 80 <= score < 90 90 <= score 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='95 confidence for each class100 score for each class Vanilla(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) 90 Label Smoothing(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='78/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) KD(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='67/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) CutMix(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) Off-LDL(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='81/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) 80 On-LDL(83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='39/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) On-LDL+(83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='62/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038) 70 score < 60 60 <= score < 70 60 70 <= score < 80 80 <= score < 90 90 <= score 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='95 confidence for each classResNet50 ResNet18 Figure 5: The top 10 high-probability classes according to the best-performing classes by train- ing methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For each method, the F1 score for the best performing class was recorded next to the class name and overall accuracy was recorded next to the method name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' CutMix: nipple (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='68), leopard (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='32) Online LDL: mud turtle (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05), library (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='19) Online LDL: nipple (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='23), leopard (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06) CutMix: mud turtle (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='31), library (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='68) Figure 6: Examples of supervision by CutMix and LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' These examples from ImageNet are label distribution and input sample pairs obtained from a teacher network after CutMix augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Vanilla Label Smoothing KD Off-LDL CutMix On-LDL On-LDL+ Figure 7: Visualization of penultimate layer’s activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The first row is the training samples of the ’beaver’, ’otter’, and ’dolphin’ classes, and the second row is the test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The third row is the training samples of the ’man’, ’woman’, and ’sunflower’ classes, and the last row is the test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' All plots are visualization of the activation of ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 5 Conclusion We observed and analyzed the effects of label smoothing, KD, and data augmentation on classification accuracy and model confidence in terms of label distribution learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Although the current approach has limitations in that the method for generating labels is relatively simple and limited to image classification, through experiments and visualization on four data sets, the LDL-based training approach can simultaneously improve model accuracy and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' As a future study, we plan to verify the utility of LDL in other tasks and analyze LDL based on theoretical backgrounds such as class-considered approaches and risk minimization [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 9 tank(95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='96) CutMix(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 2 tor E Classsunflower(94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='36) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0040 Label Smoothing(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='68) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 Probability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 Classawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='mower(95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='43 Vanilla(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 cle truck motor C trad Classbicycle(95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='92) On-LDL+(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='51) Vanilla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='008 Label Smoothing KD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='006 Probability CutMix Off-LDL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='004 On-LDL On-LDL+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='000 Classsunflower(96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='48) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0040 On-LDL(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 6 SW Classsunflower(96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0040 Off-LDL(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='86) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 Probabilit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 Classmotorcycle(94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0035 KD(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='67) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 ke cra awn_ Classsunflower(93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='94) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 LabelSmoothing(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 Classlawn mower(93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='88) Vanilla(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55） 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0025 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0000 Classpickup truck(96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='52) On-LDL+(83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='62) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='006 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='000 Classsunflower(98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0030 On-LDL(83.' 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regularization and data augmentation are class dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='03632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' [42] Gal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' & Ghahramani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' (2016) Dropout as a Bayesian approximation: Representing model uncertainty in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' of ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' [43] Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=', Tran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' & F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' (2020) What makes training multi-modal classification networks hard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' of CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Supplementary Materials Overview This supplementary material contains additional experiments and analyses not included in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 1) Experimental results with uncertainty based LDL, 2) Additional model calibration error measurement results, 3) Loss curve and training tendency, 4) Additional visualization of feature representation, 5) LDL evaluation results for each dataset according to the student/teacher configurations, 6) Pseudocode of LDL, and 7) Reliability diagrams according to each method are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 6 Uncertainty based LDL In addition to data augmentation, we evaluated uncertainty-based LDL as an online LDL method for generating labels during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Uncertainty-based LDL adopts variational inference [42] on teacher networks to generate random labels ˆz = 1 N �N i=1 σ(θt,i(x)), where N is the total number of teacher networks dropped from MCDO and σ is the softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We set the drop parameters of 11 MCDO ρ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 and N to 20 through empirical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We used the mean of the softmax output for MC dropout-based variance inference as ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We have applied another method for uncertainty-based LDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Overfitting-to-generalization-ratio (OGR) is originally designed to prevent overfitting during mutlmodal training by defining the ten- dency of the gradient of each modality during training as an adaptive loss [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We modified the OGR as per class loss difference between validation and training loss and use it as an adap- tive loss or as label distribution by normalized weighting one-hot label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The OGR is defined as OGRk = ((Lval,N+n−Ltrain,N+n)−(Lval,N −LtrainN ))2 Lval,N+n−Lval,N , where given model parameters θ at epoch N, L is average loss of each kth class over the fixed train set, and n is gap of epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We set n to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 1 shows the results obtained with each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Uncertainty-based LDL methods were also able to achieve mostly improved performance compared to vanilla models, but the improvement was small compared to on/offline LDL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Nevertheless, in almost all configurations of MCDO, a positive effect could also be achieved in model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In the case of OGR, the gain was not even greater than that of MCDO, but when OGR was applied to the LDL method, more improved results were obtained in accuracy and model correction than when the class-wise loss was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' It was confirmed that overconfidence occurs when OGR is used as a class-wise loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We speculate that the reason why MCDO and OGR are difficult to achieve as much performance improvement as the online LDL technique is that they generate the label distribution through only the change of the label without considering the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Nevertheless, uncertainty-based LDL was still able to validate its potential in accuracy and model calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 6: Classification accuracy and ECE of the uncertainty based LDL and other approaches for the CIFAR 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' MCDO [42] based LDL are marked with T and S according to the application of the teacher and student networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' OGR [42] includes two approaches: each class OGR is reflected in loss or LDL is reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Teacher ResNet200 Student ResNet18 ResNet50 ResNet200 Vanilla 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='080 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='35/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='107 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='58/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='101 Temperature Scaling [4] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='35/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='032 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='58/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='029 Label smoothing 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='085 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='24/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='042 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='42/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='037 KD (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='1,T=3) [7] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='16/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='077 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='45/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='103 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='23/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='100 CRD+KD [12] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='108 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='07/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='118 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='20/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='110 WSL [16] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='03/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='111 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='114 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='107 MCDO [42] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='03/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='010 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='104 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='23/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='110 On-LDL (MCDO-T) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='95±0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 7 Additional Model Calibration Error Measurement In addition to the ECE in the experiments, different calibration error metrics include uncertainty calibration error (UCE), negative log-likelihood (NLL), and Brier score [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Table 2 shows that for most model calibration error measurement metrics, the LDL-based method achieves the lowest error except for temperature scaling, which is simple normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In particular, the NLL and Brier scores achieve the lowest error in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' In the case of MCDO-based LDL, improved results were observed in ECE, but high errors were observed in the remaining metrics, and very high in NLL and Brier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 8 Loss Curve and Training Tendency Figure 8 shows the loss curves according to ResNet50-based vanilla, label smoothing, and online LDL methods in CIFAR100 and training trends according to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The loss curve plotted the log loss of the top 5 and bottom 5 accuracy classes to investigate the training trends for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The most notable feature in the trend of the loss curve is the largest difference between the top 5 and bottom 5 losses for vanilla training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' LDL and label smoothing suppress this tendency, and the 12 Table 7: Evaluation results according to model calibration error scores (ECE, UCE, NLL, and Brier) with different teacher (T) and student (S) configuration in CIFAR 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' T(R200)/S(R18) Score Metric acc ece uce nll Brier Vanilla 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='766 On-LDL 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='094 On-LDL+ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='092 tendency becomes more pronounced after the first drop in learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' LDL has the worst test performance at the beginning of training but has the highest accuracy after the first drop in learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' This tendency is also observed when the log loss for the top and bottom 20 classes is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' We observed the effect of label distribution learning in preventing overconfidence by class, and then we are considering an approach that can adaptively apply label distribution and data augmentation by class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The trend of this loss curve is similarly observed in previous studies [9, 10], but we were able to observe the cause more clearly in the log loss by class, and the online LDL makes a more firm contribution to the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 13 Figure 8: Log loss and accuracy plots (ResNet50) for top and bottom (5 and 20 classes) for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The upper graph plotted the top and bottom 5 classes, and the bottom graph plotted 20 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 14 80 Vanilla Label Smoothing 60 On-LDL 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 0 50 100 150 200 250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 Top20 Vanilla 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 M Bottom20 Vanilla ss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 M Top20 LS 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 Bottom20 LS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 Top20 On-LDL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 Bottom20 On-LDL 0 50 100 150 200 250 Epochs80 Vanilla Label Smoothing 60 On-LDL c 40 A 20 0 0 50 100 150 200 250 Top5 Vanilla 1 Bottom5 Vanilla ss Top5 LS 9 Bottom5 LS Top5 On-LDL 1 Bottom5 On-LDL 0 50 100 150 200 250 Epochs9 Additional Visualization of Feature Representation Figure 9 shows additional visualization of the activation of the penultimate layer of ResNet50 for ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Each plot consists of two semantically similar classes of ImageNet and one semantically different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' From the first line in Figure 9: [’green snake’,’water snake’,’grown’], [’golf ball’,’ping-pong ball’,’pot’], [’pizza’,’potpie’,’espresso’], [’leopard’,’snow leopard’,’goldfish’], [’timber wolf’,’brush wolf’,’black bear’], [’persian cat’,’siamese cat’,’mink ’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The first two classes are semantically similar, and the last class is a semantically different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Through penultimate layer visualization, it is observed that the on/offline LDL technique generates an effective representation for classifying semantically similar classes except the case of [’timber wolf’,’brush wolf’,’black bear’] (5th row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Vanilla LS KD CutMix Off-LDL On-LDL On-LDL+ Figure 9: Visualization of penultimate layer’s activation of ResNet50 for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' From the top: [’green snake’,’water snake’,’grown’], [’golf ball’,’ping-pong ball’,’pot’], [’pizza’,’potpie’,’espresso’], [’leopard’,’snow leopard’,’goldfish’], [’timber wolf’,’brush wolf’,’black bear’], [’persian cat’,’siamese cat’,’mink ’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The first two classes are semantically similar, and the last class is a semantically different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 green snake(63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='16) water snake(65.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 4 2 0 210 LDL Model Comparison Figure 10 shows a comparison plot between the proposed LDL-based best-performing model and other training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' For each plot, the x-axis is classification accuracy and the y-axis is ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Each graph shows the accuracy/ECE relationship for all student networks trained from a specific teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The CNN models with the LDL approach in all datasets are located in the lower right of the graph, which means that the LDL achieves low ECE and high classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' CIFAR100 CIFAR10 STL10 ImageNet Figure 10: Comparison of performance across specific datasets and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The x- axis is classification accuracy, and the y-axis is ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Each graph was plotted in different colors and shapes according to the training methodology, and each model name is indicated in each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 16 T:R44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12 Vanilla LS resnet20 KD esnet32 esnet110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 resnet44 Our best resnet56 esnet110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08 resneti resnet110 resnet E C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 resneti res 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 resr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 80 81 82 83 84 85 86 87 88 89 AccuracyT:R110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='07 Vanilla LS resne4et56 KD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 resnet110 Our best resnet32 resnet20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' resh eti10 resesi esne C resne et20 E resnet20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 resnet4 resi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='01 resnet2 snets 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 91 92 93 94 95 96 AccuracyT:R56 Vanilla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14 LS pehe110 KD 5f5 6 SSKD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12 resnet110 CRD+KD resnet56 resnet20 resnet44 Our best resnet5ssnet11( resne 32 esnet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 snet32 resnet32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08 re net 0 C resnet20 resnet20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 resnet20 esnet110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 resnet32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 re resnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' resnet56 resnet44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00 68 70 72 74 76 78 AccuracyT:R110 Vanilla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='14 resretdt 10 LS resnet56 resnet20 resnet44 resnet110 KD snet56 et56 SSKD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='12 resnet20 resnet56 esnet32 resnet110 CRD+KD esnet44 res*4 WSL Our best 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 resnetet3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08 C resnnet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 resnet20 resnet11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 resnet32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 res resnet3 resnet56 resnet44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00 68 70 72 74 76 78 AccuracyT:R200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='200 Vanilla LS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='175 KD CRD+KD WSL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='150 resnext2$ 4x64d Our best 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='125 reset50 t200 res t50 lesnetz snet200 C dla res 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='100 dla t18 resi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='075 resnet18 reesreex22940604d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='050 4x64d densenet121 00 rdlheltsmet200 t121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='000 76 78 80 82 84 86 AccuracyT:R152 Vanilla resnet18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='10 LS KD Our best 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='08 resnet50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='06 resnet101 C resntt re1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='04 resnet15 eshepst50 esnet50 101 52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='02 reettis she 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='00 68 70 72 74 76 78 80 Accuracy11 Implementation of On/offline LDL 1 import torch 2 import torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='nn as nn 3 import numpy as np 4 5 criterion = nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' CrossEntropyLoss () 6 for images , labels in loader: 7 # Generate data augments Images 8 images = Data_Aug(image , aug_method) 9 10 # Generate soft labels 11 if method in [’Off -LDL’, ’On -LDL’]: 12 with torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='no_grad (): 13 labels = TeacherModel(images) 14 15 logits = StudentModel(images) 16 17 # Cross entropy loss 18 loss = criterion(logits , labels) 19 20 def Data_Aug(images , aug_method , alpha =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0): 21 lam = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='beta(alpha , alpha) 22 I_x , I_y = images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='size ()[2:] 23 # shuffle minibatch 24 index = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randperm(images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='size (0)) 25 rand_image = images[index] 26 if aug_method == ’mixup ’: # MixUp Algorithm 27 images = lam * images + (1 - lam) * rand_image 28 elif aug_method == ’ricap ’: #RICAP Algorithm 29 # draw a boundary position (w,h) 30 w = int(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='round(I_x * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='beta(alpha , alpha ))) 31 h = int(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='round(I_y * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='beta(alpha , alpha ))) 32 w_ = [w, I_x -w, w, I_x -w] 33 h_ = [h, I_y -h, h, I_y -h] 34 # select and crop four images 35 cropped_images = {} 36 for k in range (4): 37 index = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randperm(images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='size (0)) 38 x_k = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randint (0, I_x - w_[k] + 1) 39 y_k = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randint (0, I_y - h_[k] + 1) 40 cropped_images [k] = images[index ][:, :, x_k:x_k + w_[k], y_k:y_k + h_[k]] 41 images = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='cat( 42 (torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='cat(( cropped_images [0], cropped_images [1]) , 2) 43 torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='cat(( cropped_images [2], cropped_images [3]) , 2)), 3) 44 elif aug_method == ’cutmix ’: # CutMix Algorithm 45 cut_w = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='int(I_x * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='sqrt (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' - lam)) # cut rate 46 cut_h = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='int(I_y * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='sqrt (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' - lam)) 47 48 cx = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randint(I_x) 49 cy = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='randint(I_y) 50 51 bbx1 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='clip(cx - cut_w // 2, 0, I_x) 52 bby1 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='clip(cy - cut_h // 2, 0, I_y) 53 bbx2 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='clip(cx + cut_w // 2, 0, I_x) 54 bby2 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='clip(cy + cut_h // 2, 0, I_y) 55 images [:, :, bbx1:bbx2 , bby1:bby2] = rand_image [:, :, bbx1:bbx2 , bby1:bby2] 56 57 return images 17 12 Reliability Diagram Reliability diagram can intuitively visualize the correlation between model confidence and inference accuracy based on a histogram [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Figure 11, 12, 13, and 14 show the reliability diagram for each dataset and approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The x-axis of the reliability diagram is a histogram of the confidence at a specific interval, and the y-axis is the expected value of the accuracy for the inferred confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' The better the model correction effect is, the more the plot values match the linear lines for the x and y axes [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Vanilla Label Smoothing KD Off-LDL (best) On-LDL (best) Figure 11: Reliability diagram for CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' Accuracy and ECE for the dataset are indicated for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' 18 T:R110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content=' S:R20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='0 Expected Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='2 ACC=92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='45 ECE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFQT4oBgHgl3EQf8DZn/content/2301.13444v1.pdf'} +page_content='035 0.' metadata={'source': 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Kalloor1 and Adar Sharon2 +1Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, +Israel +2Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794, U.S.A. +January 5, 2023 +Abstract +We discuss aspects of the quantum Lyapunov exponent λL in theories with an exactly +marginal SYK-like random interaction, where λL can be computed as a continuous +function of the interaction strength J . In 1d, we prove a conjecture from [1] which +states that at small J , λL can be found by considering a specific limit of the four- +point function in the decoupled theory. We then provide additional evidence for the 2d +version of this conjecture by discussing new examples of Lyapunov exponents which can +be computed at weak coupling. +1 +arXiv:2301.01353v1 [hep-th] 3 Jan 2023 + +Contents +1 +Introduction +3 +2 +Background +5 +2.1 +Four-point function and double-commutator . . . . . . . . . . . . . . . . . . +6 +2.2 +Chaos +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3 +The continuity conjecture +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3 +Proving the continuity conjecture in QM +12 +4 +The chiral SYK model +12 +5 +The disordered N = 2 A3 minimal model +14 +5.1 +Details of duality to free fields . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5.2 +Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +5.3 +Chaos +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +6 +Conclusions +21 +A +Superappendix +23 +B Simplification of 1d integral +28 +2 + +1 +Introduction +The quantum Lyapunov exponent λL is an intriguing and complicated observable in quantum +field theories. Of particular interest are theories where λL saturates an upper bound [2], since +this might indicate that they have semiclassical gravity duals. However, in this paper we will +be interested in the opposite limit, and would like to study how the chaos exponent behaves +as we turn on some small coupling J . The classical version of this question has been studied +extensively, and many interesting behaviors have been found for chaos at weak coupling. For +example, the KAM theorem schematically states that an integrable system deformed by a +small (integrability-breaking) deformation remains non-chaotic even for a finite but small +enough deformation. We will ask an analogous question in a quantum setting: starting with +N decoupled theories and slowly increasing the strength of a random interaction between +them, how does λL behave? +Our setup is the following. The basic building block is a conformal field theory (CFT) +C, called the “core CFT”, which contains a primary Φ. Next, we take N copies of the core +CFT, and deform it by a random interaction: +CN + Ji1...iq +� +ddxΦi1...Φiq . +(1.1) +The couplings Ji1...iq are random with Gaussian measure and variance ⟨J2 +i1...iq⟩ = (q − 1)! J 2 +Nq−1 +(with no sum over repeated indices).1 The deformation should be understood in terms of +conformal perturbation theory around N copies of the CFT C, and we will be studying these +theories in the large-N limit. We will call such theories “disordered CFTs”. +Disordered CFTs are a generalization of ideas appearing in disordered free theories, which +can be obtained by setting C to be a free field theory, and setting Φ to be the corresponding +free field. A famous example of disordered free theories is the SYK model [3, 4], and some +additional examples can be found in [5–13]. In general, the random interactions allow for +some exact computations in the IR for disordered free fields, assuming the theory flows to +an interacting CFT [3, 14, 15]. This was generalized to arbitrary core CFTs in [1, 16]. In +these theories, λL can be read off from a certain out-of-time-ordered four-point function (or +alternatively, the double-commutator) [2, 4, 17, 18]. This is a difficult computation in general, +but it is aided by the large-N limit and conformal invariance in the IR. In particular, we must +also restrict ourselves to dimensions d ≤ 2, since we will be interested in the theory at finite +temperature, but in small dimensions a conformal transformation can map such theories to +flat space. +1We emphasize that our disorder is spacetime-independent. +3 + +An especially interesting subsector of disordered CFTs can be obtained by demanding +that the interaction term in (1.1) is exactly marginal, such that J parametrizes a line of +CFTs. This behavior is not generic, but can be obtained by using supersymmetry or by +considering chiral theories, both of which we will discuss in this paper.2 Having a line of +CFTs allows us to follow interesting observables as we continuously move between different +CFTs, and in particular will allow us to study λL as a function of J . +In [1], the chaos exponent was studied in the weak-coupling limit J → 0 in disordered +CFTs where J is exactly marginal. In particular, in the examples that were discussed it was +observed the chaos exponent in the limit J → 0 is also given by the leading exponential +behavior of the double-commutator (DC) in a single core CFT. In other words, if the DC +of a single core CFT behaves at large times as exp(λ0 +Lt), then the chaos exponent of the +interacting theory in the limit J → 0 is given by +λL(J → 0) = λ0 +L . +(1.2) +It was conjectured that this result is general; we will call it the continuity conjecture. +The equality (1.2) is surprising for two main reasons. First, λ0 +L cannot be interpreted as +a chaos exponent in a single core CFT (since some form of a large-N limit is required for +this interpretation), and yet in the interacting theory it dictates the behavior of the chaos +exponent at weak coupling. Second, it is possible for λ0 +L to be negative. A negative chaos +exponent seems counter-intuitive, and indeed the result cannot be trusted in the standard +method of computing λL due to some assumptions that are required. A more precise version +of the continuity conjecture is then +λL(J → 0) = max(λ0 +L, 0) . +(1.3) +Despite this fact, if λ0 +L is negative then one can still determine that the chaos exponent of +the theory is at most zero for a finite range of values of J . As a result, it is enough to show +that in a single core CFT λ0 +L < 0 in order to find a discontinuous transition into chaos, see +figure 1. This is reminiscent of classical KAM theory. +This paper includes two main results. First we will prove the continuity conjecture (1.3) +in quantum mechanics (QM). We then discuss two examples in 2d and show that they obey +the continuity conjecture, providing further evidence for the 2d version of the conjecture. +The first example is the chiral SYK model discussed in [12], where the chaos exponent was +already found for all J . We compare the result at small J to λ0 +L and find agreement. Next +we discuss the disordered A3 minimal model. Since this theory is dual to a free CFT, we can +2Similar behavior can be obtained in QM without these assumptions [11]. +4 + +(a) +(b) +Figure 1: Two types of behaviors for the dependence of λL on the exactly +marginal interaction J : (a) continuous and (b) discontinuous. +compute all n-point functions for its primaries, and we use this result to compute the chaos +exponent at small J . Comparing the result to λ0 +L we again find agreement. +In the examples discussed in this paper, λ0 +L always turns out to be non-negative, so that +the physical picture is that of figure 1a. We will discuss ideas for how to generate cases with +a negative value and a discontinuous transition into chaos. +This paper is organized as follows. In section 2 we introduce the theories we consider in +this paper and the methods for computing the chaos exponent. We then prove the continuity +conjecture in QM in section 3. In section 4 we discuss our first 2d example, the chiral SYK +model, and show that the continuity conjecture is obeyed. In section 5 we discuss our second +example, the disordered A3 minimal model. We compute the chaos exponent at small J and +again show that the continuity conjecture is obeyed. We discuss some future directions in +section 6. +2 +Background +In this section we review the basic method discussed in [1] for obtaining the chaos exponent +λL at weak coupling for a general disordered CFT. +5 + +ΛLΛL +Jc2.1 +Four-point function and double-commutator +In disordered free theories, it is possible to write down self-consistency equations for the two- +and four-point functions of the fundamental field, which can then be solved using a conformal +ansatz when the conformal symmetry is restored in the IR. In [1] it was shown that similar +equations can be written down for general disordered CFT for the operator Φ in (1.1), which +can in principle be solved for any value of J when it is an exactly marginal deformation. +However, solving these equations requires knowledge of all n-point functions of Φ in the core +CFT, and so is very difficult in general. In this section we will review the derivation in [1], +omitting some details. +In the following we will only use the four-point function, and so we only discuss the self- +consistency equation for the four-point function and for the DC. The discussion can also be +immediately generalized to SUSY theories, in which case the diagrams discussed should be +understood as supergraphs. +We would first like to compute the connected four-point function +C = 1 +N 2 +N +� +i,j=1 +⟨ΦiΦiΦjΦj⟩conn +(2.1) +in the deformed theory (1.1). +The diagrams contributing to C have an iterative ladder +structure at large N, so that the full result for the four-point function is given by +C = +∞ +� +n=0 +KnF0 = +F0 +1 − K , +(2.2) +where F0, K are defined in figure 2. The computation of K and F0 requires knowing all +“subtracted n-point functions”, denoted by n′ +s. +These are given by the standard n-point +functions but with some specific subtractions of lower-order correlators, and are explicitly +defined in [1]. Examples of some subtracted n-point functions appear in figure 3. From these +it is in principle possible to compute the full four-point function C. One can also perform +perturbation theory in J ; at leading order only the first diagram in the expression for K +in figure 2 contributes, which is determined by the CFT four-point function and two-point +function, and takes the form +K(1, 2; 3, 4) = (q − 1)J2⟨Φ1 ¯Φ2Φ3 ¯Φ4⟩′ +sG(3, 4)q−2 , +(2.3) +with G(3, 4) the Φ propagator between the points x3, x4. +6 + +Figure 2: The kernel K and initial diagram F0 for general disordered CFTs. +Red lines denote full propagators G, and black dots denote insertions of the +disorder interaction, with q − 2 red propagators between each pair. +Figure 3: Examples of correlation functions n′ +s. Dashed lines corresponds to +external points, while solid lines are connected via Σ’s in figure 2. +Next we discuss the computation of the double-commutator, defined as +WR(t1, t2) = 1 +N 2 +N +� +i,j=1 +⟨[Φi(β/2), Φj(β/2 + it2)][Φi(0), Φj(it1)]⟩ += lim +ε→0 +1 +N 2 +N +� +i,j=1 +⟨(Φi (ε) − Φi (−ε)) (Φi (β/2 + ε) − Φi (β/2 − ε)) +· Φj (it1) Φj (β/2 + it2)⟩ . +(2.4) +We have suppressed the spatial coordinates, since they will be unimportant, and we have +kept the (real and imaginary) time coordinates. By ⟨...⟩ we mean the Euclidean time-ordered +thermal trace. Note that (2.4) is just a combination of analytically-continued Euclidean four- +point functions on the cylinder, which in a d = 2 CFT is an analytically-continued flat-space +correlator (2.1). +The diagrams contributing to the double-commutator also obey an iterative ladder struc- +ture. The corresponding kernel KR and initial diagram F0,R have the same diagrammatics as +7 + +K= +十. +Fo=Z +2 +Y +2 +7 +6 +4 +2Figure 4: The complex time contour chosen for the computation of the DC. +Red dots denote insertion points of operators. We call the two excursions +from the real axis the “left rail” and the “right rail”. +the four-point function (appearing in figure 2), but the correlators are analytically-continued +versions of the ones appearing above. The details appear in [1], and we will write down +explicitly only the leading contribution in J , which comes from the four-point function: +KR(t1, t2, t3, t4) = J 2 +� +∆O (it1) ∆O +�β +2 + it2 +� +O (it3) O +�β +2 + it4 +��′ +s +Gq−2 +lr,∆(3, 4) + O(J 4) . +(2.5) +Here we have denoted ∆O(z) = O(z + ε) − O(z − ε) where we eventually take the limit +ϵ → 0. The integration range in principle for all points is over the complex time contour +appearing in figure 4, although various cancellations as we take ϵ → 0 lead to the result +above. In particular, we have used the expression for the thermal two-point function for a +scalar operator of dimension ∆ between points from different “rails” (see figure 4): +Glr,∆(1, 2) = +1 +� +4 cosh +� t12−x12 +2 +� +cosh +� t12+x12 +2 +��∆ . +(2.6) +2.2 +Chaos +Having written down the ladder diagrams which contribute to the DC, we can now compute +the chaos exponent. In principle, this is done by computing the DC explicitly, and then taking +the large-time limit t1, t2 = t → ∞, which should lead to exponentially growing behavior, +WR(t1, t2) ∼ exp +�λL +2 (t1 + t2) +� +f(t1 − t2) . +(2.7) +where from now on we set β = 2π. However, computing the full DC is difficult, and instead +the existence of an iterative ladder structure offers us a shortcut. The ladder structure leads +8 + +1 +β +it +1 +- +2 +β +B +E +E +3+ +2 +2to the self-consistency equation +WR = F0,R + KRWR . +(2.8) +At large times, we assume that the F0,R term is negligible, and so WR obeys the equation +WR = KRWR , +(2.9) +which is just an eigenvalue equation for KR. As a result, the exponentially-growing solution +for WR must be an eigenfunction of the retarded kernel KR with eigenvalue 1. Thus, the +chaos exponent is found by guessing solutions of the form (2.7) and finding their eigenvalue +kR(λL) under KR. The largest λL for which kR(λL) = 1 is the chaos exponent. +The precise form of the eigenvalue is constrained due to conformal invariance, and takes +the form of a two-point function on the cylinder. In 2d we have the ansatz +Wλ(1, 2) = +exp +� +− h+˜h +2 (t1 + t2) + h−˜h +2 (x1 + x2) +� +(2 cosh +� t12−x12 +2 +� +)∆−h(2 cosh +� t12+x12 +2 +� +)∆−˜h . +(2.10) +In general we require h = − λ +2 + ip, ˜h = − λ +2 − ip for real λ, p, but in practice in the examples +appearing here the maximal chaos exponent will have p = 0. +Finding the chaos exponent now amounts to computing the eigenvalue of the eigenfunction +Wλ under KR. Since the theories we consider are conformally invariant for any value of J , +we can perform this computation in a perturbative expansion in J . Generally the eigenvalue +is given by +kR(λ, J ) = +� +d2x3d2x4KR · W +W +, +(2.11) +and plugging in the leading-order result for KR we find that in 2d and at leading order in J , +kR(λ, J ) =J 2 +� +d2x3d2x4 ⟨∆O (it1) ∆O (β/2 + it2) O (it3) O (β/2 + it4)⟩′ +s +· +Glr,∆+ λ +2 (3, 4) +Glr,∆+ λ +2 (1, 2) exp +�λ +2(t3 + t4 − t1 − t2) +� +· Glr,2∆(q−2)(3, 4) + O(J4) +=J 2 +4 +exp +� +− λ +2(t1 + t2) +� +Glr,λ/2(1, 2) +� ∞ +u1 du3 +� u2 +−∞ du4 +u +2+ λ +2 +34 +� ∞ +v1 dv3 +� v2 +−∞ dv4 +v +2+ λ +2 +34 +GR(χ, ¯χ) + O(J 4) . +(2.12) +With χ, ¯χ the conformal cross-ratios. Here we have performed the change of variables +u3 = ex3−t3 , +v3 = e−x3−t3 , +u4 = −ex4−t4 , +v4 = −e−x4−t4 . +(2.13) +9 + +GR is the retarded normalized 4-point of the undeformed CFT at J = 0, +GR(χ, ¯χ) = ⟨[O(β/2 + it2), O(β/2 + it4)][O(it1), O(it3)]⟩′ +s +Glr,∆(1, 2)Glr,∆(3, 4) += +lim +ε1,ε2→0 G++ − G+− − G−+ + G−− +(2.14) +with the normalized four-point G±1,±2 = G(u1e±iε1, v1e±iε1, u2e±iε2, v1e±iε2, u3, v3, u4, v4). +2.2.1 +Consistency of the perturbative expansion +We now discuss perturbative corrections to the chaos exponent which are subleading in the +J → 0 limit. The computation of the chaos exponent in perturbation theory in J requires +several assumptions. Expanding the eigenvalues of the retarded kernel, we expect to see +kR(λ, J ) = J 2f2(λ) + J 4f4(λ) + ... , +(2.15) +First, we would like to understand when it is enough to find leading term f2(λ) discussed +above in order to find the leading value of the chaos exponent in the limit J → 0. The chaos +exponent is found by setting kR = 1, and so if we keep only the J 2 term, λL is found by +analyzing at which values of λ the function f2(λ) diverges as 1/J 2. The largest such λ can +then be identified with the chaos exponent. In order for this procedure to be consistent it is +enough to require two conditions on the higher-order terms: +1. If the largest value of λ at which f2 diverges is λ0, then all other fn diverge at values +λ ≤ λ0. +2. If f2 diverges as +1 +(λ−λ0)α, then other fn diverge as +1 +(λ−λ0)βn for βn ≤ nα/2. +In this case the leading value of λL is indeed λ0. All examples discussed in [1] were shown to +obey these conditions, and we will show that the examples discussed here also obey them). +Next, it is natural to ask when it is consistent to perform a perturbative expansion around +λ0 in the limit of small J in order to obtain the chaos exponent in a series in J of the form +λL = λ0 + J 2λ1 + ... +(2.16) +This requires a strict inequality in the second condition above, i.e. we require +βn < nα/2 . +(2.17) +To see why this is the case, assume an expansion of the form +kR = +� +an +J 2n +λ2n , +(2.18) +10 + +so that α = 2 and βn = n = nα/2 and the inequality is exactly saturated. As a result, the +expansion is not in terms of J , but in terms of J /λ. We can now try to compute λL in +perturbation theory. Write +λL = λ0 + J 2λ1 + ... +(2.19) +Plugging in this expansion, we immediately learn that the leading order is given by λ0 = 0, +as discussed above. However, in order to find the value of the subleading term λ1, we must +take into account all of the terms in the expansion, since all terms they are all of the same +order in J . As a result, while we can still compute λ0 in such theories, we cannot perform +perturbation theory around it without knowing kR completely (or at least to all orders in +some double-scaling limit). So a perturbative calculation of λL beyond the leading order +using this method fails. We will see this happen in the chiral SYK example discussed in +section 4. +On the other hand, for any theory where there is a strict inequality βn < nα/2, a pertur- +bative expansion in J should be possible. This is the result in e.g. the generalized free fields +examples in [1]. +2.3 +The continuity conjecture +We now discuss the behavior of the chaos exponent as J → 0. +Note that in equation +(2.12), the prefactor J 2 vanishes in this limit, and so in order for us to be able to solve for +kR(λL, J ) = 1, we must find values of λ for which the integral in (2.12) diverges as 1/J 2. The +chaos exponent in this limit is then given by the largest value of λ for which this divergence +occurs. Assuming that the retarded four-point function of the theory at a single core CFT +behaves at large times t1, t2 as3 +GR(t1, t2) ∼ exp +� +λ0 +L(t1 + t2)/2 +� +, +(2.20) +it is easily seen that the divergence occurs precisely at λ = λ0 +L. As a result, it was conjectured +in [1] that the chaos exponent in the limit J → 0 is given precisely by the leading exponential +behavior of the double-commutator in the free theory at J = 0. Thus the computation of +the leading behavior of λL is particularly simple, and is a property of a single core CFT.4 +Several examples were discussed in [1] which were shown to obey this conjecture. +3In d ≤ 2 this is related to the Regge limit. +4Note that λ0 +L does not have any interpretation in terms of a chaos exponent in a single core CFT, since +this requires some form of a large-N limit or another weak-coupling expansion. +11 + +3 +Proving the continuity conjecture in QM +The continuity conjecture discussed in 2.3 relates the chaos exponent in disordered CFTs at +small J to the late-time behavior of the DC in a single core CFT. It states that assuming +that GR defined in (2.14) (or equivalently, the double-commutator) behaves at large times +as exp(λ0 +Lt), then the chaos exponent as J → 0 approaches λ0 +L. We will now prove this +statement under the assumption that perturbation theory in J is valid for the eigenvalues of +the retarded kernel, so that the leading order is obtained by considering just the contribution +of the four-point function to the retarded kernel, see section 2.2. +Consider the leading contribution to kR, which comes from the four-point function dia- +gram in figure 2. In 1d this contribution simplifies to (see Appendix B for a derivation): +kR = +1 +|z12|2∆ +� ∞ +z1 +dz3 +� z2 +−∞ +dz4 +GR(χ) +|z34|2+λ . +(3.1) +Here, we performed the change of variables z = e−t on the “left rail” (points 1, 3) and z = −e−t +on the “right rail” (points 2, 4), see e.g. [5] for details. The conformal cross-ratio is +χ = z12z34 +z13z24 +. +(3.2) +Next we perform another change of coordinates to the coordinates κ, χ, where κ is defined +as z3 = κ/χ. The integral becomes is +kR(λ) = +� 0 +−∞ +dχGR(χ) +χ1−λ +� χ−1 +−∞ +dκ +κ1+λ(κ − χ)1+λ(κ − (χ − 1))−λ . +(3.3) +The chaos exponent λL is now given by the largest value of λ for which the integral diverges +in the limit t3, t4 → −∞, which in these coordinates corresponds to χ → 0. Note that the κ +integral is finite as long as λ > −1 (corresponding to λ0 +L > −1). At χ = 0 the κ integral is +smooth; it does not vanish or diverge. Therefore we can expand GR(χ) around χ = 0 inside +the integral. Let us denote the leading order term by GR(χ) = c0χ−λ0 +L + .... Note that since +at large times we have χ ∼ e−t, we can identify λ0 +L with the rate of growth of the DC of a +single core CFT as in (2.20). Plugging in this expansion, we see that the integral converges +only for λ > λ0 +L, which means that the chaos exponent at small J is λ0 +L. We have thus proved +the continuity conjecture in 1d. +4 +The chiral SYK model +The chiral SYK model was introduced in [12] (see also [19]). The theory is a disordered free +theory in 2d, where the core CFT is a free chiral Majorana fermion. The theory was shown +12 + +to have a line of fixed points characterized by J , so that the chaos exponent can be found +as a function of J . Since the theory is chiral, instead of the standard chaos exponent it is +more interesting to consider the velocity-dependent chaos exponent, given by considering a +large time and large distance limit with the ratio v = x/t kept constant. The corresponding +chaos exponent is denoted λv. It was found that the chaos exponent always starts at zero +as J → 0, and rises as we increase J . In particular, choosing v such that λv is maximal, +one finds that at infinite J the chaos exponent λv saturates the bound on chaos [2]. We +will be interested in the weak-coupling limit, where J is close to zero. We will show that +the continuity conjecture is obeyed for the velocity-dependent chaos exponent, and discuss +corrections in J . +The velocity-dependent chaos exponent at weak coupling can be easily extracted from +Appendix B of [12] (see equation (B.6)), and in the limit J → 0 one finds +λv(J ) = +� +� +� +2π +β η(v − 1) + O(J ), +u− < v < u+ +0, +else +(4.1) +where u± = 1 ± J +2π and +η = +√ +3 +√ +1 − J 2 +1 − v +� +J 2 − (1 − v)2 . +(4.2) +Note that since u− < v < u+, η is finite in the limit J → 0. +Since the theory is chiral, it is not surprising that the chaos exponent vanishes outside +of a cone around the speed of light. Taking the strict J → 0 limit, we find that λv(J → 0) +vanishes trivially unless v = 1 (since u− = u+ = 1), but at v = 1 it turns out to vanish as +well. Thus we find λv(J → 0) = 0. +We would like to compare this to λ0 +L, which describes the large-time behavior of the DC +in a single copy of the free core CFT. The DC in the core CFT is given by GR(13)GR(24), +where GR(ij) is the retarded propagator between spacetime points i, j and is given by +GR(t, x) = +1 +β√u+u− +Θ +� +t − u−1 ++ x +� +Θ +� +u−1 +− x − t +� +� +sinh +� +π +β +� +t − u−1 ++ x +�� +sinh +� +π +β +� +u−1 +− x − t +�� . +(4.3) +Next we take the large-time limit where t = t1 = t2 and x = x1 = x2 are large while v = x/t +is kept constant. We find (up to unimportant constants) +DC ∝ +� +� +� +exp +� +− π +β(u−1 +− − u−1 ++ )vt +� +, +u− < v < u+ +0, +else +(4.4) +13 + +Plugging in J = 0 we find as a result that λ0 +v = 0 always. Thus we have found +λ0 +v = λv(J → 0) = 0 , +(4.5) +and so the continuity conjecture is obeyed. +We can also try to understand subleading corrections to the chaos exponent as discussed +in section 2.2.1. The eigenvalues of the chiral SYK model were computed in [12], and we can +expand the result in J : +kR = 3J 2 +λ2 − 6J 3 +λ3 + O +� +J 4� +. +(4.6) +Expanding to higher orders one finds α = 2 and βn = n = nα/2 in the notation of section +2.2.1. +As a result, the expansion is not in terms of J , but in terms of J /λ, and so a +perturbative computation of λL will fail at higher orders in J . +5 +The disordered N = 2 A3 minimal model +We now discuss the case where the core CFT is the N = 2 supersymmetric A3 minimal +model. This model can be constructed using a single chiral superfield X with superpotential +W = X4 . +(5.1) +The model has central charge c = 3/2 and its spectrum includes a chiral primary of dimension +1/4 which we will call Φ. This theory can be identified with the theory of a free boson H an +a free fermion χ, which combine into a single free N = 1 chiral multiplet [20]. The core CFT +thus has a free field representation. +The disordered A3 minimal model with q = 4 can then be constructed as +(A3)N + +� +i1̸=...̸=i4 +Ji1...i4 +� +d2xd2θΦi1...Φi4 . +(5.2) +In particular, this deformation is marginal. In fact, as discussed in [1], it can be shown +using standard arguments [21–23] that any realization of this theory is a CFT, so that the +interaction is exactly marginal (even without averaging). +In order to compute λL we need to know n-point functions of Φ. Since the A3 minimal +model has a free field realization in terms of N = 1 superfields, we should be able to identify +the components of Φ with products of operators from the free boson and free fermion CFT. +In practice, the various components will be mapped to products of vertex operators and +fermionic twist fields, whose n-point functions are known. +Plugging the results into the +14 + +retarded kernel, we will be able to read off λL. Following the discussion above, we will focus +on the four-point function which gives the leading contribution to λL at small J . +Our notations appear in appendix A. In particular, we use lightcone coordinates x± so +that a two-point function takes the form (x+x−)∆ ≡ |x|2∆. +5.1 +Details of duality to free fields +The free field representation of the A3 minimal model consists of a free Majorana fermion χ +and a free compact boson H at the self-dual radius R = 1/ +√ +2 (see for instance [20] or [24]). +The SUSY algebra is generated by the operators +G± = χ± exp +� +i +√ +2H± +� +, +¯G± = χ± exp +� +−i +√ +2H± +� +, +j(R) +± += +i +√ +2∂±H , +(5.3) +where ± stand for the left/right moving parts of the fields and operators.5 +The fundamental superfield Φ has scaling dimensions +� 1 +8, 1 +8 +� +, and there are five superpri- +maries in the NS sector: Φ, Φ2, ¯Φ, ¯Φ2, ¯ΦΦ. Their bottom components map to the following +free theory operators: +φ = σ exp +� +i H +2 +√ +2 +� +, +¯φ = σ exp +� +−i H +2 +√ +2 +� +, +φ2 = exp +� +i H +√ +2 +� +, +¯φ2 = exp +� +−i H +√ +2 +� +, +¯φφ = ϵ = χ+χ− , +(5.4) +where σ and µ (which will be of use later), with (h, ¯h) = (1/16, 1/16), are the twist fields of +the fermion theory. The other components of these superfields may be worked out via free +field OPEs (see Appendix A). For convenience, we list in Table 1 the components of the basic +superfield: +Φ(y+, y−) = φ(y+, y−) + θ+ψ+(y+, y−) + θ−ψ−(y+, y−) + θ+θ−F(y+, y−) +(5.5) +along with their dimensions and R-charges. +5The theory actually has N = 3 SUSY, but we will only need the N = 2 subalgebra above for our +purposes. +15 + +Table 1: The operators in the multiplet of Φ in the A3 minimal model. +O +Ofree +(h, ¯h) +qR +φ +σ ei H +2 +√ +2 +� 1 +8, 1 +8 +� +� 1 +4, 1 +4 +� +ψ+ +µ ei +−3H++H− +2 +√ +2 +� 5 +8, 1 +8 +� +� +− 3 +4, 1 +4 +� +ψ− +µ ei +H+−3H− +2 +√ +2 +� 1 +8, 5 +8 +� +� 1 +4, − 3 +4 +� +F +σ e−i 3H +2 +√ +2 +� 5 +8, 5 +8 +� +� +− 3 +4, − 3 +4 +� +5.2 +Correlation functions +Having identified the components of the superfield Φ with operators from free field theories, +we can now compute all n-point functions of the field Φ. +5.2.1 +Two-point function +Superconformal symmetry fixes the form of the two-point function to be +⟨Φ¯Φ⟩ = +1 +|⟨12⟩|2∆ +(5.6) +where the relevant value for our theory is ∆ = 1/4. By expanding the result in the superspace +coordinates on both sides, we can identify two-point functions of the various components of +Φ, which allows us to set their normalization. We find +⟨φ¯φ⟩ = +1 +|x12|2∆ , +⟨F ¯F⟩ = +4∆2 +|x12|2(∆+1) , +⟨ψ+ ¯ψ+⟩ = − +2∆ +|x12|2∆x+ +12 +, +⟨ψ− ¯ψ−⟩ = − +2∆ +|x12|2∆x− +12 +. +(5.7) +5.2.2 +Four-point function +We move on to the computation of the four-point function. Superconformal symmetry fixes +its form to be +⟨Φ(x1)¯Φ(x2)Φ(x3)¯Φ(x4)⟩ = +1 +|⟨12⟩⟨34⟩| +1 +2 f(χ+ +s , χ− +s ) , +(5.8) +where f is an arbitrary function of the superconformal cross ratios +χ± +s = ⟨12⟩±⟨34⟩± +⟨14⟩±⟨32⟩± . +(5.9) +16 + +Note that the bottom component of χ± +s is the usual non-supersymmetric cross-ratio χ± = +x± +12x± +34 +x± +14x± +32. +We start by calculating the bottom component of this four point function. Using the +identification (5.4), this is equivalent to computing the product of a four-point function of +vertex operators and of twist operators σ. The results are well known, and we find +⟨φ(x1)¯φ(x2)φ(x3)¯φ(x4)⟩ = +���� +1 +x12x34 +���� +1 +2 +1 +√ +2 +� +1 + |χ| + |χ − 1| . +(5.10) +Since there is a single superconformal cross-ratio of each chirality, it is simple to uplift this +result to the full supermultiplet; it suffices to replace χ± → χ± +S and the prefactor of +��� +1 +x12x34 +��� +1 +2 +with its supersymmetric analog +1 +|⟨12⟩⟨34⟩| +1 +2 . The result is thus +⟨Φ(x1)¯Φ(x2)Φ(x3)¯Φ(x4)⟩ = +1 +|⟨12⟩⟨34⟩| +1 +2 +1 +√ +2 +� +1 + |χs| + |χs − 1| . +(5.11) +As a consistency check, we have checked that expanding both sides in the superspace +coordinates gives the expected result for four-point functions of other components of Φ. +5.3 +Chaos +We can now compute the chaos exponent using the procedure outlined in section 2.2. This +requires diagonalizing the retarded kernel. We perform this diagonalization by first writing +down the kernel for the standard four-point function, and then performing the analytic con- +tinuation required to obtain the retarded kernel. In the following we will focus on the bottom +component of all four-point functions, assuming that they produce that leading behavior at +long times, as was the case in similar models [1, 5, 6]. +5.3.1 +The chaos exponent at weak coupling +We start with the eigenvalues of the standard kernel. The eigenvalues of the bosonic kernel +are given by +k(h, J ) = +� +d2X3d2 ¯X4K · W +W +, +(5.12) +where d2X = d2xd2θ and d2 ¯X = d2xd2¯θ. At leading order in J the kernel K is given by the +leading (super-)diagram in figure 2 and W is given by (2.10). Plugging in the form of the +17 + +four-point function (5.11) we find6 +� +KW = +� +d2X3d2 ¯X4⟨Φ1 ¯Φ2Φ3 ¯Φ4⟩ +1 +|⟨34⟩|3/2−2h = +1 +√ +2|⟨12⟩|1/2 +� +d2X3d2 ¯X4 +� +1 + |χs| + |χs − 1| +|⟨34⟩|2−2h +. +(5.13) +We will focus on the bosonic part of this integral. Using +⟨12⟩± = x± +12 , +⟨34⟩± = x± +34 − 2θ± +3 ¯θ± +4 , +χ± +s = x± +12(x± +34 − 2θ± +3 ¯θ± +4 ) +x± +14x± +32 += χ±(1 − 2θ± +3 ¯θ± +4 +x± +34 +) . +(5.14) +we can perform the Grassman integrals to obtain +k = +1 +|z12|1/2 +� +d2x3d2x4 +(x− +34)h−2(x+ +34)h−2 +4 +√ +2 +I(χ±) +(5.15) +where +I = +1 +(|χ − 1| + |χ| + 1)3/2 +�|χ| (4h (|χ − 1| − χ+ + 1) − 2|χ − 1| + 3χ+ − 2) + χ− (7|χ| − 2χ+ + 4) +|χ − 1| ++ χ− (4h (2χ+|χ − 1| + 2χ+|χ| − |χ − 1| − 2|χ| + χ+ − 1) − 6χ+ (|χ − 1| + |χ|) + 4|χ − 1|) +|χ − 1| +−4(h − 1) (|χ − 1| + |χ| + 1) +�|χ| − χ+ (|χ − 1| + |χ|) +χ+ − 1 +− 4(h − 1) (|χ − 1| + |χ| + 1) +�� +. +(5.16) +We can now use analytic continuation to find the eigenvalues of the retarded kernel at +finite temperature. More precisely, first we need to go to the cylinder, the do the analytic +continuation. The mapping to the cylinder is given by [5] +u = ex−t, +v = e−x−t, +(left rail) +u = −ex−t, +v = −e−x−t, +(right rail) +(5.17) +with the rails identified in figure 4. The analytic continuation is done by taking the following +times: +τ1 = β/2 ± ϵ1 + it1, +τ2 = ±ϵ2 + it2, +τ3 = β/2 + it3, +τ4 = it4 . +(5.18) +The procedure is then to compute the four-point function for each of the four choices of +signs for the ϵ’s, and then add them up where each term gets a sign which is positive if both +epsilons have the same sign and negative otherwise. This gives the double-commutator +� +[Φi (β/2 + it1) , Φj (β/2 + it3)][Φi (it2) , Φj (it4)] +� +(5.19) +6We are ignoring the subtractions here. They are necessary to compute the eigenvalues k, but will not +contribute to the eigenvalues of the retarded kernel which are our main interest. +18 + +which can be written as +� +(Φi (β/2 + ϵ1 + it1) − Φi (β/2 − ϵ1 + it1)) +� +Φi (ϵ2 + it2) − Φi (−ϵ2 + it2) +� +Φj (β/2 + it3) Φj (it4) +� +, +(5.20) +where we eventually take the limit ϵi → 0. Note that for most combinations of τ’s, the +ϵ-dependence drops out in this limit. For example, in the combination τ1 − τ4 the real part is +always dominated by β and not by the ϵ term, and so the sign of ϵ does not affect the result +when we take ϵ → 0. As a result, χ±(ϵ) = χ± does not depend on ϵ: +χ± = sinh x12±iτ12 +2 +sinh x34±iτ34 +2 +sinh x14±iτ14 +2 +sinh x32±iτ32 +2 +. +(5.21) +However, |χ − 1| does depend on ϵ. Following this procedure for |χ − 1| we find that the +analytic continuation takes +|χ − 1| = +� +(χ− − 1)(χ+ − 1) → −4 +� +|(χ− − 1)(χ+ − 1)| , +(5.22) +assuming t13 > |x13| and t24 > |x24|, and the expression vanishes otherwise. +Up to overall factors which will not be important, the integral then becomes +kR = +� +(KW)R = +� +du3dv3du4dv4 +1 +u2−h +34 v2−h +34 +IR +(5.23) +where IR is given by taking I and subtracting I but where each |χ−1| is replaced by −|χ−1|. +In these new variables, we have +χ = u12u34 +u14u32 +, +¯χ = v12v34 +v14v32 +, +(5.24) +and the integration region is u3 > u1, u2 > u4, v3 > v1, v2 > v4. +As discussed in 2.2, the chaos exponent as J → 0 is found by looking for divergences, +which should appear at large |ui|, |vi|; the chaos exponent is λ0 = −2h∗ where h∗ is the value +for which the integral diverges. We can find this analytically, since we expect the divergence +to come from taking large u3, v3, u4, v4 (and they are all of the same order of magnitude), +which means we are taking χ, ¯χ → 0. In this limit the integrand behaves as +1 +u2−h +34 v2−h +34 +(5.25) +which diverges for h ≥ 0. So the chaos exponent at weak coupling is +λL(J → 0) = 0 . +(5.26) +19 + +5.3.2 +Contributions from higher n-point functions +As discussed in section 2.2, we must make sure that contributions to the kernel from higher +n-point functions don’t diverge at lower values of λ than λL(J → 0) = 0, otherwise the +approximations made above are invalid. We know all of the higher n-point functions since +we have mapped the theory to a free theory, and so it is a matter of plugging the results into +the kernel.7 +As an example, we consider the contribution from the bottom component of the 2n-point +function for any n to the kernel in figure 2. Using the free field realization, we find that any +2n-point function of the bottom component φ of the superfield Φ takes the form +⟨φ(x1)...φ(xn)¯φ(y1)...¯φ(yn)⟩ = +1 +2n/2 +� +� +� +� +� +� +� +ϵx +i =±1, ϵy +i =±1, +� ϵx +i +ϵy +i =0 +� +i